x Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 3-1 Example 3-1: Solution

Save this PDF as:
 WORD  PNG  TXT  JPG

Size: px
Start display at page:

Download "x Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 3-1 Example 3-1: Solution"

Transcription

1 Chapter 3 umerical Descriptive Measures 3.1 Measures of Central Tendency for Ungrouped Data 3. Measures of Dispersion for Ungrouped Data 3.3 Mean, Variance, and Standard Deviation for Grouped Data 3.4 Use of Standard Deviation 3.5 Measures of Position 3. Box-and-Whisker Plot Mean 3.1 Measures of Central Tendency for Ungrouped Data The mean for ungrouped data is obtained by dividing the sum of all values by the number of values in the data set Mean for population data: x Median Mean for sample data: The median is the value of the middle term in a data set that has been ranked in increasing order Mode The mode is the value that occurs with the highest frequency in a data set Relationships among the Mean, Median, and Mode x x n 1 Example 3-1 Example 3-1: Solution Table 3.1 lists the total cash donations (rounded to millions of dollars) given by eight U.S. companies during the year 010 x x1 x x 3 x 4 x 5 x x 7 x x 111 x $139. 5million n 8 Thus, these eight companies donated an average of $139.5 million in 010 for charitable purposes. Find the mean of cash donations made by these eight companies 3 4 1

2 Example 3- The following are the ages (in years) of all eight employees of a small company: Find the mean age of these employees The population mean is x years Thus, the mean age of all eight employees of this company is 45.5 years, or 45 years and 3 months Example 3-3 Table 3. lists the total number of homes lost to foreclosure in seven states during 010 ote that the number of homes foreclosed in California is very large compared to those in the other six states. Hence, it is an outlier. Show how the inclusion of this outlier affects the value of the mean 5 Examples 3-3 Solution Median If we do not include the number of homes foreclosed in California (the outlier), the mean of the number of foreclosed homes in six states is Mean with out the outlier 49,73 01,9 0,35 10,84 40,911 18,038 1,848 33,1 ow, to see the impact of the outlier on the value of the mean, we include the number of homes foreclosed in California and find the mean number of homes foreclosed in the seven states. Mean with the outlier 173,175 49,73 0,35 10,84 40,911 18,038 1, ,871 53, How to find the median Rank the data set in increasing order. Find the middle term. The value of this term is the median. Example of weight lost: 10, 5, 19, 8, 3 Rank the data: 3, 5, 8, 10, 19 Find the median: 3, 5, 8, 10, 19 What if there are numbers: 3, 5, 8, 10, 13, 19 The median gives the center of a histogram, with half the data values to the left of the median and half to the right of the median. The advantage of using the median as a measure of central tendency is that it is not influenced by outliers. Consequently, the median is preferred over the mean as a measure of central tendency for data sets that contain outliers. 8

3 Example 3-4 Refer to the data on the number of homes foreclosed in seven states given in Table 3. of Example 3.3. Data: 173,175 49,73 0,35 10,84 40,911 18,038 1,848 Find the median for these data. Solution Rank the data: 10,84 18,038 0,35 40,911 49,73 1, ,175 Locate the middle term 10,84 18,038 0,35 40,911 49,73 1, ,175 Thus, the median number of homes foreclosed in these seven states was 40,911 in 010. Example 3-5 Table 3.3 gives the total compensations (in millions of dollars) for the year 010 of the 1 highest-paid CEOs of U.S. companies. Data: 3.9,.9, 8., 84.5, 1., 8.0, 7.1, 5., 3., 70.1,.5, 1.7 Find the median Solution Rank the data Locate the middle term Thus, the median is ( ) / = Mode The mode is the value that occurs with the highest frequency in a data set Example 3- The following data give the speeds (in miles per hour) of eight cars that were stopped on I-95 for speeding violations Find the mode. A major shortcoming of the mode is that a data set may have none or may have more than one mode, whereas it will have only one mean and only one median. Unimodal: A data set with only one mode. Bimodal: A data set with two modes. Multimodal: A data set with more than two modes. Relationships among Mean, Median, & Mode For a symmetric histogram and frequency curve with one peak, the values of the mean, median, and mode are identical, and they lie at the center of the distribution

4 Relationships among Mean, Median, & Mode For a histogram and a frequency curve skewed to the right, the value of the mean is the largest, that of the mode is the smallest, and the value of the median lies between these two. (otice that the mode always occurs at the peak point.) The value of the mean is the largest in this case because it is sensitive to outliers that occur in the right tail. These outliers pull the mean to the right. Relationships among Mean, Median, & Mode If a histogram and a distribution curve are skewed to the left, the value of the mean is the smallest and that of the mode is the largest, with the value of the median lying between these two. In this case, the outliers in the left tail pull the mean to the left Measures of Dispersion for Ungrouped Data Range Range = maximum value minimum value Variance Deviation from mean ( & ) Kind of average of squared deviation Population variance Sample variance Why divided by n-1, instead of by n for sample variance Standard deviation = square root of variance Why do we need both variance and standard deviation? Population parameters and sample statistics Range = largest value smallest value Example of two students test scores in a class A: ; B: Disadvantages The range, like the mean has the disadvantage of being influenced by outliers. Consequently, the range is not a good measure of dispersion to use for a data set that contains outliers. Its calculation is based on two values only: the largest and the smallest. All other values in a data set are ignored when calculating the range

5 Variance and Standard Deviation The standard deviation is the most used measure of dispersion. The value of the standard deviation tells how closely the values of a data set are clustered around the mean. In general, a lower value of the standard deviation for a data set indicates that the values of that data set are spread over a relatively smaller range around the mean. In contrast, a large value of the standard deviation for a data set indicates that the values of that data set are spread over a relatively large range around the mean. The standard deviation is obtained by taking the positive square root of the variance The variance calculated for population data is denoted by σ² (read as sigma squared), and the variance calculated for sample data is denoted by s². The standard deviation calculated for population data is denoted by σ, and the standard deviation calculated for sample data is denoted by s. 17 Calculation of Variance Get deviations from the mean & then square the deviations Kind of average of the squared deviations x (x - ) (x - ) = 5 5 = = 0 0 = = 0 0 = = 0 0 = = -5 (-5) = 5 18 Book s formula Way Basic Formulas for the Variance and Standard Deviation x Short-cut Formulas x x x and x x x x and s s n 1 n 1 x x x and x x x and s s n n 1 n n 1 Example 3-1 Until about 009, airline passengers were not charged for checked baggage. Around 009, however, many U.S. airlines started charging a fee for bags. According to the Bureau of Transportation Statistics, U.S. airlines collected more than $3 billion in baggage fee revenue in 010. The following table lists the baggage fee revenues of six U.S. airlines for the year 010. (ote that Delta s revenue reflects a merger with orthwest. Also note that since then United and Continental have merged; and American filed for bankruptcy and may merge with another airline.) Find the variance and standard deviation for these data next slide

6 Formula Solution of Example 3-1 Two Observations 1. The values of the variance and the standard deviation are never negative.. The measurement units of variance are always the square of the measurement units of the original data. The standard deviation, not the variance, is the most used measure of dispersion. s x,854 x 1,74,098 n n 1 1 1,74,098 1,357,55.7 s 77, , Example 3-13 Following are the 011 earnings (in thousands of dollars) before taxes for all six employees of a small company Calculate the variance and standard deviation for these data. ( x) x , Population Parameters and Sample Statistics A numerical measure such as the mean, median, mode, range, variance, or standard deviation calculated for a population data set is called a population parameter, or simply a parameter. A summary measure calculated for a sample data set is called a sample statistic, or simply a statistic. Slight difference in formula of population variance and sample variance $

7 3.3 Mean, Variance & Standard Deviation for Grouped Data Grouped data in frequency table Midpoint of each group (class) as approximate data point, and frequency as the number of such approximate points Then follow the conventional definitions for mean, variance, and standard deviation Mean for Grouped Data Mean for population data mf mf x n Mean for sample data where m is the midpoint and f is the frequency of a class Variance and Standard Deviation for Grouped Data Example 3-14 Table 3.8 gives the frequency distribution of the daily commuting times (in minutes) from home to work for all 5 employees of a company. Calculate the mean of the daily commuting times. Equivalent to data set: 5, 5, 5, 5; 15, 15, 15, 15, 15, 15, 15, 15, 15; 5, 5, 5, 5, 5, 5; 35, 35, 35, 35; 45, 45 Mean = 535 / 5 = Example 3-15 Table 3.10 gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Calculate the mean of orders Equivalent to data set: Mean = 83 / 50 = 1.4 Variance and Standard Deviation for Grouped Data Basic formulas f m f m x and s n 1 Short-cut formulas ( ) mf mf m f m f and s n n 1 where σ² is the population variance, s² is the sample variance, and m is the midpoint of a class. In either case, the standard deviation is obtained by taking the positive square root of the variance 7 8 7

8 Example 3-1 Example 3-1: Solution The following data, reproduced from Table 3.8 of Example 3-14, give the frequency distribution of the daily commuting times (in minutes) from home to work for all 5 employees of a company. Calculate the variance and standard deviation. m ( f mf ) (535 ) 14, minutes Thus, the standard deviation of the daily commuting times for these employees is 11. minutes. Recall the equivalent data set 9 30 Examples of Variance for Grouped Data Example 3-17: Table 3.10 gives the frequency distribution of the number of orders received each day during the past 50 days at the office of a mail-order company. Looks like we have to count on the formulas not really Mean is integer Computer packages: Excel, Minitab, etc. Illustrative example modified from example 3.1 data Time f m mf m f (m-mean) f 0 to < to < to < to < to < Sum Formula way Concept way σ = 3800/4 = Use of Standard Deviation Chebyshev s Theorem For any number k greater than 1, at least (1 1/k²) of the data values lie within k standard deviations of the mean. Empirical Rule For a bell shaped distribution approximately 8% of the observations lie within one standard deviation of the mean 95% of the observations lie within two standard deviations of the mean 99.7% of the observations lie within three standard deviations of the mean 3 8

9 Chebyshev s Theorem Example 3-18 The average systolic blood pressure for 4000 women who were screened for high blood pressure was found to be 187 with a standard deviation of. Using Chebyshev s theorem, find at least what percentage of women in this group have a systolic blood pressure between 143 and 31. μ = 187 and σ = k = (3), area 75% (89%) 143 μ = k = 44/ = The percentage is at least k () 4.75 or 75% Illustration of the Empirical Rule Example 3-19 The age distribution of a sample of 5000 persons is bellshaped with a mean of 40 years and a standard deviation of 1 years. Determine the approximate percentage of people who are 1 to 4 years old. 1. Compare the numbers with the mean. Check how many standard deviations the number is away from the mean

10 Example 3-19? The age distribution of a sample of 5000 persons is bellshaped with a mean of 40 years and a standard deviation of 1 years. Determine the approximate percentage of people who are 8 to 4 years old. Determine the approximate percentage of people who are 1 to 5 years old. Determine the approximate percentage of people who are 5 years or old. Determine the approximate percentage of people who are 8 years or young. 3.5 Measures of Position Quartiles: Q1, Q, Q3 Divide the ranked data into four equal parts Interquartile range = Q3 Q1 Box-and-Whisker Plot Percentile: P1, P,, P99 Divide the ranked data into 100 equal parts Q1 = P5, Q = P50 = median, Q3 = P75 Standard score Use for comparison of different subjects Quartiles and Percentiles Example 3-0 Table 3.3 in Example 3-5 gave the total compensations (in millions of dollars) for the year 010 of the 1 highest-paid CEOs of U.S. companies. That table is reproduced on the next slide. Find the values of the three quartiles. Where does the total compensation of Michael D. White (CEO of DirecTV) fall in relation to these quartiles? Find the interquartile range

11 Rank the data first Example 3-0: Solution By looking at the position of $3.9 million (total compensation of Michael D. White, CEO of DirecTV), we can state that this value lies in the bottom 75% of the 010 total compensation. This value falls between the second and third quartiles. IQR = Interquartile range = Q3 Q1 = = $7.45 million Example 3-1 The following are the ages (in years) of nine employees of an insurance company: Find the values of the three quartiles. Where does the age of 8 years fall in relation to the ages of the employees? Find the interquartile range. Solution Rank the data: Q1 = 30.5, Q = 37, Q3 = 49 The age of 8 falls in the lowest 5% of the ages IQR = Interquartile range = Q3 Q1 = = 18.5 Inclusive or exclusive in the calculations see Excel 41 4 Percentiles and Percentile Rank 3. Box-and-Whisker Plot Calculating Percentiles The (approximate) value of the k th percentile, denoted by P k, is kn P k Value of the th term in a ranked data set 100 where k denotes the number of the percentile and n represents the sample size 0 th percentile Ex 3-: Finding Percentile Rank of a Value umber of valuesless than xi Percentilerank of xi 100 Totalnumber of valuesin thedata set Examples 3- & 3-3 Different packages might give you different answers Excel Minitab Five-umber Summary Min, Q1, Q=Median, Q3, Max A plot that shows the center, spread, and skewness of a data set. It is constructed by drawing a box and two whiskers that use the median, the first quartile, the third quartile, and the smallest and the largest values in the data set between the lower and the upper inner fences. Formal BW plot Also used to detect potential outliers Box = Q1, Q, Q3 Whisker Inner fences: 1.5 times of IQR away from Q1(Q3) Outer fences: 3 times of IQR away from Q1(Q3)

12 Example 3-4 The following data are the incomes (in thousands of dollars) for a sample of 1 households Construct a box-and-whisker plot for these data. Step 1: rank the data and calculate Q1, Q, Q3 & IQR and draw box Step : calculate 1.5 x IQR and find the lower (upper) inner fence = Q1 (Q3) (+) 1.5 x IQR go beyond the box Step 3: locate the smallest (largest) values within the two inner fences Step 4: draw whiskers Step 5: uses and misuses Detecting outliers is a challenging problem Example 3-4 Q1 = 77, Q = 87, Q3 = 101, IQR = x IQR = 1.5 x 4 = 3 Lower inner fence = Q1 3 = 41, upper = Q3 + 3 = 137 Two whiskers: 9 (smallest value > 41) & 11 (largest < 137) 45 4 Standard Score Technology Instruction Standard score defined as Use for comparison An example: compare Mike s height (78 ins) with Rebecca s height (7 ins) BA players: average height = 9 ins & s =.8 ins Mike Jordan s std height is WBA players: average height = 3. ins & s =.5 ins Rebecca Lobo s std height is Who has advantage in height when they play? TI 84 / 83 plus Minitab Excel

Numerical Measures of Central Tendency

Numerical Measures of Central Tendency Numerical Measures of Central Tendency Often, it is useful to have special numbers which summarize characteristics of a data set These numbers are called descriptive statistics or summary statistics. A

More information

We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students:

We will use the following data sets to illustrate measures of center. DATA SET 1 The following are test scores from a class of 20 students: MODE The mode of the sample is the value of the variable having the greatest frequency. Example: Obtain the mode for Data Set 1 77 For a grouped frequency distribution, the modal class is the class having

More information

Chapter 3: Data Description Numerical Methods

Chapter 3: Data Description Numerical Methods Chapter 3: Data Description Numerical Methods Learning Objectives Upon successful completion of Chapter 3, you will be able to: Summarize data using measures of central tendency, such as the mean, median,

More information

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes

A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes A frequency distribution is a table used to describe a data set. A frequency table lists intervals or ranges of data values called data classes together with the number of data values from the set that

More information

Describing Data. We find the position of the central observation using the formula: position number =

Describing Data. We find the position of the central observation using the formula: position number = HOSP 1207 (Business Stats) Learning Centre Describing Data This worksheet focuses on describing data through measuring its central tendency and variability. These measurements will give us an idea of what

More information

PROPERTIES OF MEAN, MEDIAN

PROPERTIES OF MEAN, MEDIAN PROPERTIES OF MEAN, MEDIAN In the last class quantitative and numerical variables bar charts, histograms(in recitation) Mean, Median Suppose the data set is {30, 40, 60, 80, 90, 120} X = 70, median = 70

More information

Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010

Data Mining Part 2. Data Understanding and Preparation 2.1 Data Understanding Spring 2010 Data Mining Part 2. and Preparation 2.1 Spring 2010 Instructor: Dr. Masoud Yaghini Introduction Outline Introduction Measuring the Central Tendency Measuring the Dispersion of Data Graphic Displays References

More information

13.2 Measures of Central Tendency

13.2 Measures of Central Tendency 13.2 Measures of Central Tendency Measures of Central Tendency For a given set of numbers, it may be desirable to have a single number to serve as a kind of representative value around which all the numbers

More information

Statistics Chapter 3 Averages and Variations

Statistics Chapter 3 Averages and Variations Statistics Chapter 3 Averages and Variations Measures of Central Tendency Average a measure of the center value or central tendency of a distribution of values. Three types of average: Mode Median Mean

More information

Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.)

Center: Finding the Median. Median. Spread: Home on the Range. Center: Finding the Median (cont.) Center: Finding the Median When we think of a typical value, we usually look for the center of the distribution. For a unimodal, symmetric distribution, it s easy to find the center it s just the center

More information

Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives

Chapter 3 Descriptive Statistics: Numerical Measures. Learning objectives Chapter 3 Descriptive Statistics: Numerical Measures Slide 1 Learning objectives 1. Single variable Part I (Basic) 1.1. How to calculate and use the measures of location 1.. How to calculate and use the

More information

Descriptive Statistics. Frequency Distributions and Their Graphs 2.1. Frequency Distributions. Chapter 2

Descriptive Statistics. Frequency Distributions and Their Graphs 2.1. Frequency Distributions. Chapter 2 Chapter Descriptive Statistics.1 Frequency Distributions and Their Graphs Frequency Distributions A frequency distribution is a table that shows classes or intervals of data with a count of the number

More information

Numerical Summaries. Chapter 2. Mean or Average. Median (M) Basic Practice of Statistics - 3rd Edition

Numerical Summaries. Chapter 2. Mean or Average. Median (M) Basic Practice of Statistics - 3rd Edition Numerical Summaries Chapter 2 Describing Distributions with Numbers Center of the data mean median Variation range quartiles (interquartile range) variance standard deviation BPS - 5th Ed. Chapter 2 1

More information

Chapter 2: Exploring Data with Graphs and Numerical Summaries. Graphical Measures- Graphs are used to describe the shape of a data set.

Chapter 2: Exploring Data with Graphs and Numerical Summaries. Graphical Measures- Graphs are used to describe the shape of a data set. Page 1 of 16 Chapter 2: Exploring Data with Graphs and Numerical Summaries Graphical Measures- Graphs are used to describe the shape of a data set. Section 1: Types of Variables In general, variable can

More information

32 Measures of Central Tendency and Dispersion

32 Measures of Central Tendency and Dispersion 32 Measures of Central Tendency and Dispersion In this section we discuss two important aspects of data which are its center and its spread. The mean, median, and the mode are measures of central tendency

More information

3: Summary Statistics

3: Summary Statistics 3: Summary Statistics Notation Let s start by introducing some notation. Consider the following small data set: 4 5 30 50 8 7 4 5 The symbol n represents the sample size (n = 0). The capital letter X denotes

More information

3.2 Measures of Spread

3.2 Measures of Spread 3.2 Measures of Spread In some data sets the observations are close together, while in others they are more spread out. In addition to measures of the center, it's often important to measure the spread

More information

10-3 Measures of Central Tendency and Variation

10-3 Measures of Central Tendency and Variation 10-3 Measures of Central Tendency and Variation So far, we have discussed some graphical methods of data description. Now, we will investigate how statements of central tendency and variation can be used.

More information

Lecture 1: Review and Exploratory Data Analysis (EDA)

Lecture 1: Review and Exploratory Data Analysis (EDA) Lecture 1: Review and Exploratory Data Analysis (EDA) Sandy Eckel seckel@jhsph.edu Department of Biostatistics, The Johns Hopkins University, Baltimore USA 21 April 2008 1 / 40 Course Information I Course

More information

Lesson 4 Measures of Central Tendency

Lesson 4 Measures of Central Tendency Outline Measures of a distribution s shape -modality and skewness -the normal distribution Measures of central tendency -mean, median, and mode Skewness and Central Tendency Lesson 4 Measures of Central

More information

Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004

Histogram. Graphs, and measures of central tendency and spread. Alternative: density (or relative frequency ) plot /13/2004 Graphs, and measures of central tendency and spread 9.07 9/13/004 Histogram If discrete or categorical, bars don t touch. If continuous, can touch, should if there are lots of bins. Sum of bin heights

More information

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI

STATS8: Introduction to Biostatistics. Data Exploration. Babak Shahbaba Department of Statistics, UCI STATS8: Introduction to Biostatistics Data Exploration Babak Shahbaba Department of Statistics, UCI Introduction After clearly defining the scientific problem, selecting a set of representative members

More information

Chapter 2. Objectives. Tabulate Qualitative Data. Frequency Table. Descriptive Statistics: Organizing, Displaying and Summarizing Data.

Chapter 2. Objectives. Tabulate Qualitative Data. Frequency Table. Descriptive Statistics: Organizing, Displaying and Summarizing Data. Objectives Chapter Descriptive Statistics: Organizing, Displaying and Summarizing Data Student should be able to Organize data Tabulate data into frequency/relative frequency tables Display data graphically

More information

1 Measures for location and dispersion of a sample

1 Measures for location and dispersion of a sample Statistical Geophysics WS 2008/09 7..2008 Christian Heumann und Helmut Küchenhoff Measures for location and dispersion of a sample Measures for location and dispersion of a sample In the following: Variable

More information

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table

2.0 Lesson Plan. Answer Questions. Summary Statistics. Histograms. The Normal Distribution. Using the Standard Normal Table 2.0 Lesson Plan Answer Questions 1 Summary Statistics Histograms The Normal Distribution Using the Standard Normal Table 2. Summary Statistics Given a collection of data, one needs to find representations

More information

Chapter 7 What to do when you have the data

Chapter 7 What to do when you have the data Chapter 7 What to do when you have the data We saw in the previous chapters how to collect data. We will spend the rest of this course looking at how to analyse the data that we have collected. Stem and

More information

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56

2. Describing Data. We consider 1. Graphical methods 2. Numerical methods 1 / 56 2. Describing Data We consider 1. Graphical methods 2. Numerical methods 1 / 56 General Use of Graphical and Numerical Methods Graphical methods can be used to visually and qualitatively present data and

More information

Introduction to Descriptive Statistics

Introduction to Descriptive Statistics Mathematics Learning Centre Introduction to Descriptive Statistics Jackie Nicholas c 1999 University of Sydney Acknowledgements Parts of this booklet were previously published in a booklet of the same

More information

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion

Descriptive Statistics. Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Descriptive Statistics Purpose of descriptive statistics Frequency distributions Measures of central tendency Measures of dispersion Statistics as a Tool for LIS Research Importance of statistics in research

More information

The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median

The right edge of the box is the third quartile, Q 3, which is the median of the data values above the median. Maximum Median CONDENSED LESSON 2.1 Box Plots In this lesson you will create and interpret box plots for sets of data use the interquartile range (IQR) to identify potential outliers and graph them on a modified box

More information

Descriptive Statistics

Descriptive Statistics Y520 Robert S Michael Goal: Learn to calculate indicators and construct graphs that summarize and describe a large quantity of values. Using the textbook readings and other resources listed on the web

More information

Session 1.6 Measures of Central Tendency

Session 1.6 Measures of Central Tendency Session 1.6 Measures of Central Tendency Measures of location (Indices of central tendency) These indices locate the center of the frequency distribution curve. The mode, median, and mean are three indices

More information

1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10. Bellwork 3-23-15 Simplify each expression.

1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10. Bellwork 3-23-15 Simplify each expression. Bellwork 3-23-15 Simplify each expression. 1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10 1 Objectives Find measures of central tendency and measures of variation for statistical data.

More information

Section 3.1 Measures of Central Tendency: Mode, Median, and Mean

Section 3.1 Measures of Central Tendency: Mode, Median, and Mean Section 3.1 Measures of Central Tendency: Mode, Median, and Mean One number can be used to describe the entire sample or population. Such a number is called an average. There are many ways to compute averages,

More information

Frequency Distributions

Frequency Distributions Displaying Data Frequency Distributions After collecting data, the first task for a researcher is to organize and summarize the data to get a general overview of the results. Remember, this is the goal

More information

Homework 3. Part 1. Name: Score: / null

Homework 3. Part 1. Name: Score: / null Name: Score: / Homework 3 Part 1 null 1 For the following sample of scores, the standard deviation is. Scores: 7, 2, 4, 6, 4, 7, 3, 7 Answer Key: 2 2 For any set of data, the sum of the deviation scores

More information

CHINHOYI UNIVERSITY OF TECHNOLOGY

CHINHOYI UNIVERSITY OF TECHNOLOGY CHINHOYI UNIVERSITY OF TECHNOLOGY SCHOOL OF NATURAL SCIENCES AND MATHEMATICS DEPARTMENT OF MATHEMATICS MEASURES OF CENTRAL TENDENCY AND DISPERSION INTRODUCTION From the previous unit, the Graphical displays

More information

F. Farrokhyar, MPhil, PhD, PDoc

F. Farrokhyar, MPhil, PhD, PDoc Learning objectives Descriptive Statistics F. Farrokhyar, MPhil, PhD, PDoc To recognize different types of variables To learn how to appropriately explore your data How to display data using graphs How

More information

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences

Introduction to Statistics for Psychology. Quantitative Methods for Human Sciences Introduction to Statistics for Psychology and Quantitative Methods for Human Sciences Jonathan Marchini Course Information There is website devoted to the course at http://www.stats.ox.ac.uk/ marchini/phs.html

More information

Summarizing Data: Measures of Variation

Summarizing Data: Measures of Variation Summarizing Data: Measures of Variation One aspect of most sets of data is that the values are not all alike; indeed, the extent to which they are unalike, or vary among themselves, is of basic importance

More information

Means, standard deviations and. and standard errors

Means, standard deviations and. and standard errors CHAPTER 4 Means, standard deviations and standard errors 4.1 Introduction Change of units 4.2 Mean, median and mode Coefficient of variation 4.3 Measures of variation 4.4 Calculating the mean and standard

More information

! x sum of the entries

! x sum of the entries 3.1 Measures of Central Tendency (Page 1 of 16) 3.1 Measures of Central Tendency Mean, Median and Mode! x sum of the entries a. mean, x = = n number of entries Example 1 Find the mean of 26, 18, 12, 31,

More information

Chapter 3: Central Tendency

Chapter 3: Central Tendency Chapter 3: Central Tendency Central Tendency In general terms, central tendency is a statistical measure that determines a single value that accurately describes the center of the distribution and represents

More information

Ch. 3.1 # 3, 4, 7, 30, 31, 32

Ch. 3.1 # 3, 4, 7, 30, 31, 32 Math Elementary Statistics: A Brief Version, 5/e Bluman Ch. 3. # 3, 4,, 30, 3, 3 Find (a) the mean, (b) the median, (c) the mode, and (d) the midrange. 3) High Temperatures The reported high temperatures

More information

GCSE HIGHER Statistics Key Facts

GCSE HIGHER Statistics Key Facts GCSE HIGHER Statistics Key Facts Collecting Data When writing questions for questionnaires, always ensure that: 1. the question is worded so that it will allow the recipient to give you the information

More information

not to be republished NCERT Measures of Central Tendency

not to be republished NCERT Measures of Central Tendency You have learnt in previous chapter that organising and presenting data makes them comprehensible. It facilitates data processing. A number of statistical techniques are used to analyse the data. In this

More information

CHAPTER 3 CENTRAL TENDENCY ANALYSES

CHAPTER 3 CENTRAL TENDENCY ANALYSES CHAPTER 3 CENTRAL TENDENCY ANALYSES The next concept in the sequential statistical steps approach is calculating measures of central tendency. Measures of central tendency represent some of the most simple

More information

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY

Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY Biostatistics: DESCRIPTIVE STATISTICS: 2, VARIABILITY 1. Introduction Besides arriving at an appropriate expression of an average or consensus value for observations of a population, it is important to

More information

STATISTICS FOR PSYCH MATH REVIEW GUIDE

STATISTICS FOR PSYCH MATH REVIEW GUIDE STATISTICS FOR PSYCH MATH REVIEW GUIDE ORDER OF OPERATIONS Although remembering the order of operations as BEDMAS may seem simple, it is definitely worth reviewing in a new context such as statistics formulae.

More information

4. DESCRIPTIVE STATISTICS. Measures of Central Tendency (Location) Sample Mean

4. DESCRIPTIVE STATISTICS. Measures of Central Tendency (Location) Sample Mean 4. DESCRIPTIVE STATISTICS Descriptive Statistics is a body of techniques for summarizing and presenting the essential information in a data set. Eg: Here are daily high temperatures for Jan 6, 29 in U.S.

More information

Data Analysis: Describing Data - Descriptive Statistics

Data Analysis: Describing Data - Descriptive Statistics WHAT IT IS Return to Table of ontents Descriptive statistics include the numbers, tables, charts, and graphs used to describe, organize, summarize, and present raw data. Descriptive statistics are most

More information

Shape of Data Distributions

Shape of Data Distributions Lesson 13 Main Idea Describe a data distribution by its center, spread, and overall shape. Relate the choice of center and spread to the shape of the distribution. New Vocabulary distribution symmetric

More information

Desciptive Statistics Qualitative data Quantitative data Graphical methods Numerical methods

Desciptive Statistics Qualitative data Quantitative data Graphical methods Numerical methods Desciptive Statistics Qualitative data Quantitative data Graphical methods Numerical methods Qualitative data Data are classified in categories Non numerical (although may be numerically codified) Elements

More information

Numerical Summarization of Data OPRE 6301

Numerical Summarization of Data OPRE 6301 Numerical Summarization of Data OPRE 6301 Motivation... In the previous session, we used graphical techniques to describe data. For example: While this histogram provides useful insight, other interesting

More information

Stats Review Chapters 3-4

Stats Review Chapters 3-4 Stats Review Chapters 3-4 Created by Teri Johnson Math Coordinator, Mary Stangler Center for Academic Success Examples are taken from Statistics 4 E by Michael Sullivan, III And the corresponding Test

More information

DATA HANDLING (3) Overview. Measures of dispersion (or spread) about the mean (ungrouped data) Lesson. Learning Outcomes and Assessment Standards

DATA HANDLING (3) Overview. Measures of dispersion (or spread) about the mean (ungrouped data) Lesson. Learning Outcomes and Assessment Standards 42 DATA HANDLING (3) Learning Outcomes and Assessment Standards Learning Outcome 4: Data handling and probability Assessment Standard AS 1(a) Calculate and represent measures of central tendency and dispersion

More information

Descriptive Data Summarization

Descriptive Data Summarization Descriptive Data Summarization (Understanding Data) First: Some data preprocessing problems... 1 Missing Values The approach of the problem of missing values adopted in SQL is based on nulls and three-valued

More information

Variance and Standard Deviation. Variance = ( X X mean ) 2. Symbols. Created 2007 By Michael Worthington Elizabeth City State University

Variance and Standard Deviation. Variance = ( X X mean ) 2. Symbols. Created 2007 By Michael Worthington Elizabeth City State University Variance and Standard Deviation Created 2 By Michael Worthington Elizabeth City State University Variance = ( mean ) 2 The mean ( average) is between the largest and the least observations Subtracting

More information

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination

Descriptive Statistics. Understanding Data: Categorical Variables. Descriptive Statistics. Dataset: Shellfish Contamination Descriptive Statistics Understanding Data: Dataset: Shellfish Contamination Location Year Species Species2 Method Metals Cadmium (mg kg - ) Chromium (mg kg - ) Copper (mg kg - ) Lead (mg kg - ) Mercury

More information

1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics)

1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics) 1.5 NUMERICAL REPRESENTATION OF DATA (Sample Statistics) As well as displaying data graphically we will often wish to summarise it numerically particularly if we wish to compare two or more data sets.

More information

Math Chapter 3 review

Math Chapter 3 review Math 116 - Chapter 3 review Name Find the mean for the given sample data. Unless otherwise specified, round your answer to one more decimal place than that used for the observations. 1) Bill kept track

More information

4. Introduction to Statistics

4. Introduction to Statistics Statistics for Engineers 4-1 4. Introduction to Statistics Descriptive Statistics Types of data A variate or random variable is a quantity or attribute whose value may vary from one unit of investigation

More information

Exploratory data analysis (Chapter 2) Fall 2011

Exploratory data analysis (Chapter 2) Fall 2011 Exploratory data analysis (Chapter 2) Fall 2011 Data Examples Example 1: Survey Data 1 Data collected from a Stat 371 class in Fall 2005 2 They answered questions about their: gender, major, year in school,

More information

( ) ( ) Central Tendency. Central Tendency

( ) ( ) Central Tendency. Central Tendency 1 Central Tendency CENTRAL TENDENCY: A statistical measure that identifies a single score that is most typical or representative of the entire group Usually, a value that reflects the middle of the distribution

More information

Each exam covers lectures from since the previous exam and up to the exam date.

Each exam covers lectures from since the previous exam and up to the exam date. Sociology 301 Exam Review Liying Luo 03.22 Exam Review: Logistics Exams must be taken at the scheduled date and time unless 1. You provide verifiable documents of unforeseen illness or family emergency,

More information

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses.

DESCRIPTIVE STATISTICS. The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE STATISTICS The purpose of statistics is to condense raw data to make it easier to answer specific questions; test hypotheses. DESCRIPTIVE VS. INFERENTIAL STATISTICS Descriptive To organize,

More information

Data Exploration Data Visualization

Data Exploration Data Visualization Data Exploration Data Visualization What is data exploration? A preliminary exploration of the data to better understand its characteristics. Key motivations of data exploration include Helping to select

More information

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 (b) 1

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 2 (b) 1 Unit 2 Review Name Use the given frequency distribution to find the (a) class width. (b) class midpoints of the first class. (c) class boundaries of the first class. 1) Miles (per day) 1-2 9 3-4 22 5-6

More information

MCQ S OF MEASURES OF CENTRAL TENDENCY

MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ S OF MEASURES OF CENTRAL TENDENCY MCQ No 3.1 Any measure indicating the centre of a set of data, arranged in an increasing or decreasing order of magnitude, is called a measure of: (a) Skewness (b)

More information

Frequency distributions, central tendency & variability. Displaying data

Frequency distributions, central tendency & variability. Displaying data Frequency distributions, central tendency & variability Displaying data Software SPSS Excel/Numbers/Google sheets Social Science Statistics website (socscistatistics.com) Creating and SPSS file Open the

More information

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives.

5/31/2013. 6.1 Normal Distributions. Normal Distributions. Chapter 6. Distribution. The Normal Distribution. Outline. Objectives. The Normal Distribution C H 6A P T E R The Normal Distribution Outline 6 1 6 2 Applications of the Normal Distribution 6 3 The Central Limit Theorem 6 4 The Normal Approximation to the Binomial Distribution

More information

Multiple Choice Questions Descriptive Statistics - Summary Statistics

Multiple Choice Questions Descriptive Statistics - Summary Statistics Multiple Choice Questions Descriptive Statistics - Summary Statistics 1. Last year a small statistical consulting company paid each of its five statistical clerks $22,000, two statistical analysts $50,000

More information

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs

Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Types of Variables Chapter 1: Looking at Data Section 1.1: Displaying Distributions with Graphs Quantitative (numerical)variables: take numerical values for which arithmetic operations make sense (addition/averaging)

More information

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number

1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number 1) Write the following as an algebraic expression using x as the variable: Triple a number subtracted from the number A. 3(x - x) B. x 3 x C. 3x - x D. x - 3x 2) Write the following as an algebraic expression

More information

MEAN 34 + 31 + 37 + 44 + 38 + 34 + 42 + 34 + 43 + 41 = 378 MEDIAN

MEAN 34 + 31 + 37 + 44 + 38 + 34 + 42 + 34 + 43 + 41 = 378 MEDIAN MEASURES OF CENTRAL TENDENCY MEASURES OF CENTRAL TENDENCY The measures of central tendency are numbers that locate the center of a set of data. The three most common measures of center are mean, median

More information

Chapter 6. The Standard Deviation as a Ruler and the Normal Model. Copyright 2012, 2008, 2005 Pearson Education, Inc.

Chapter 6. The Standard Deviation as a Ruler and the Normal Model. Copyright 2012, 2008, 2005 Pearson Education, Inc. Chapter 6 The Standard Deviation as a Ruler and the Normal Model Copyright 2012, 2008, 2005 Pearson Education, Inc. The Standard Deviation as a Ruler The trick in comparing very different-looking values

More information

Measures of Central Tendency and Variability: Summarizing your Data for Others

Measures of Central Tendency and Variability: Summarizing your Data for Others Measures of Central Tendency and Variability: Summarizing your Data for Others 1 I. Measures of Central Tendency: -Allow us to summarize an entire data set with a single value (the midpoint). 1. Mode :

More information

BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I

BNG 202 Biomechanics Lab. Descriptive statistics and probability distributions I BNG 202 Biomechanics Lab Descriptive statistics and probability distributions I Overview The overall goal of this short course in statistics is to provide an introduction to descriptive and inferential

More information

Lesson 3 Measures of Central Location and Dispersion

Lesson 3 Measures of Central Location and Dispersion Lesson 3 Measures of Central Location and Dispersion As epidemiologists, we use a variety of methods to summarize data. In Lesson 2, you learned about frequency distributions, ratios, proportions, and

More information

Topic 9 ~ Measures of Spread

Topic 9 ~ Measures of Spread AP Statistics Topic 9 ~ Measures of Spread Activity 9 : Baseball Lineups The table to the right contains data on the ages of the two teams involved in game of the 200 National League Division Series. Is

More information

STP 226 Example EXAM #1 (from chapters 1-3, 5 and 6)

STP 226 Example EXAM #1 (from chapters 1-3, 5 and 6) STP 226 Example EXAM #1 (from chapters 1-3, 5 and 6) Instructor: ELA JACKIEWICZ Student's name (PRINT): Class time: Honor Statement: I have neither given nor received information regarding this exam, and

More information

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1.

Def: The standard normal distribution is a normal probability distribution that has a mean of 0 and a standard deviation of 1. Lecture 6: Chapter 6: Normal Probability Distributions A normal distribution is a continuous probability distribution for a random variable x. The graph of a normal distribution is called the normal curve.

More information

Module 4: Data Exploration

Module 4: Data Exploration Module 4: Data Exploration Now that you have your data downloaded from the Streams Project database, the detective work can begin! Before computing any advanced statistics, we will first use descriptive

More information

In this module, we will cover different approaches used to summarize test scores.

In this module, we will cover different approaches used to summarize test scores. In this module, we will cover different approaches used to summarize test scores. 1 You will learn how to use different quantitative measures to describe and summarize test scores and examine groups of

More information

The Chi-Square Distributions

The Chi-Square Distributions MATH 183 The Chi-Square Distributions Dr. Neal, WKU The chi-square distributions can be used in statistics to analyze the standard deviation " of a normally distributed measurement and to test the goodness

More information

Classify the data as either discrete or continuous. 2) An athlete runs 100 meters in 10.5 seconds. 2) A) Discrete B) Continuous

Classify the data as either discrete or continuous. 2) An athlete runs 100 meters in 10.5 seconds. 2) A) Discrete B) Continuous Chapter 2 Overview Name MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. Classify as categorical or qualitative data. 1) A survey of autos parked in

More information

MAT 142 College Mathematics Module #3

MAT 142 College Mathematics Module #3 MAT 142 College Mathematics Module #3 Statistics Terri Miller Spring 2009 revised March 24, 2009 1.1. Basic Terms. 1. Population, Sample, and Data A population is the set of all objects under study, a

More information

Report of for Chapter 2 pretest

Report of for Chapter 2 pretest Report of for Chapter 2 pretest Exam: Chapter 2 pretest Category: Organizing and Graphing Data 1. "For our study of driving habits, we recorded the speed of every fifth vehicle on Drury Lane. Nearly every

More information

Statistical Concepts and Market Return

Statistical Concepts and Market Return Statistical Concepts and Market Return 2014 Level I Quantitative Methods IFT Notes for the CFA exam Contents 1. Introduction... 2 2. Some Fundamental Concepts... 2 3. Summarizing Data Using Frequency Distributions...

More information

EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck!

EXAM #1 (Example) Instructor: Ela Jackiewicz. Relax and good luck! STP 231 EXAM #1 (Example) Instructor: Ela Jackiewicz Honor Statement: I have neither given nor received information regarding this exam, and I will not do so until all exams have been graded and returned.

More information

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I)

STAT 155 Introductory Statistics. Lecture 5: Density Curves and Normal Distributions (I) The UNIVERSITY of NORTH CAROLINA at CHAPEL HILL STAT 155 Introductory Statistics Lecture 5: Density Curves and Normal Distributions (I) 9/12/06 Lecture 5 1 A problem about Standard Deviation A variable

More information

Unit 4: Statistics Measures of Central Tendency & Measures of Dispersion

Unit 4: Statistics Measures of Central Tendency & Measures of Dispersion Unit 4: Statistics Measures of Central Tendency & Measures of Dispersion 1 Measures of Central Tendency a measure that tells us where the middle of a bunch of data lies most common are Mean, Median, and

More information

Numerical descriptive measures

Numerical descriptive measures 1/8/014 AGSC 30 Statistial Methods Numerial desriptive measures Data representation 1. Measures o entral tendeny e.g., mean, mode, median, midrange. e.g., range, variane, standard deviation 3. Measures

More information

CH.6 Random Sampling and Descriptive Statistics

CH.6 Random Sampling and Descriptive Statistics CH.6 Random Sampling and Descriptive Statistics Population vs Sample Random sampling Numerical summaries : sample mean, sample variance, sample range Stem-and-Leaf Diagrams Median, quartiles, percentiles,

More information

Central Tendency. n Measures of Central Tendency: n Mean. n Median. n Mode

Central Tendency. n Measures of Central Tendency: n Mean. n Median. n Mode Central Tendency Central Tendency n A single summary score that best describes the central location of an entire distribution of scores. n Measures of Central Tendency: n Mean n The sum of all scores divided

More information

MEI Statistics 1. Exploring data. Section 1: Introduction. Looking at data

MEI Statistics 1. Exploring data. Section 1: Introduction. Looking at data MEI Statistics Exploring data Section : Introduction Notes and Examples These notes have sub-sections on: Looking at data Stem-and-leaf diagrams Types of data Measures of central tendency Comparison of

More information

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction

CA200 Quantitative Analysis for Business Decisions. File name: CA200_Section_04A_StatisticsIntroduction CA200 Quantitative Analysis for Business Decisions File name: CA200_Section_04A_StatisticsIntroduction Table of Contents 4. Introduction to Statistics... 1 4.1 Overview... 3 4.2 Discrete or continuous

More information

Descriptive Statistics and Measurement Scales

Descriptive Statistics and Measurement Scales Descriptive Statistics 1 Descriptive Statistics and Measurement Scales Descriptive statistics are used to describe the basic features of the data in a study. They provide simple summaries about the sample

More information

CHAPTER 3 AVERAGES AND VARIATION

CHAPTER 3 AVERAGES AND VARIATION CHAPTER 3 AVERAGES AND VARIATION ONE-VARIABLE STATISTICS (SECTIONS 3.1 AND 3.2 OF UNDERSTANDABLE STATISTICS) The TI-83 Plus and TI-84 Plus graphing calculators support many of the common descriptive measures

More information

Measures of Center Section 3-2 Definitions Mean (Arithmetic Mean)

Measures of Center Section 3-2 Definitions Mean (Arithmetic Mean) Measures of Center Section 3-1 Mean (Arithmetic Mean) AVERAGE the number obtained by adding the values and dividing the total by the number of values 1 Mean as a Balance Point 3 Mean as a Balance Point

More information