13.2 Measures of Central Tendency


 Felix Baldwin
 1 years ago
 Views:
Transcription
1 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 in the set tend to cluster, a kind of middle number or a measure of central tendency. Three such measures are discussed in this section. Mean The mean (more properly called the arithmetic mean) of a set of data items is found by adding up all the items and then dividing the sum by the number of items. (The mean is what most people associate with the word average. ) The mean of a sample is denoted (read x bar ), while the mean of a complete population is denoted (the lower case Greek letter mu). The mean of n data items x 1, x 2,, x n, is given by the formula We use the symbol for summation, (the Greek letter sigma): x x n x x1 x2... xn Ex 1) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mean number of siblings for the ten students. Ex 2) Researchers at the University of Maryland collected body temperature readings from a sample of adults, and eight of those temperatures are listed below (in degrees Fahrenheit). Does the mean of this sample equal 98.6, which is commonly believed to be the mean body temperature of adults? Weighted Mean x f w. f The weighted mean of n numbers x 1, x 2,, x n, that are weighted by the respective factors f 1, f 2,, f n is given by the formula above. In a common system for finding a gradepoint average, an A is assigned 4 points, with 3 points for a B, 2 for C, and 1 for D. Find the gradepoint average by multiplying the number of units for a course and the number assigned to each grade, and then adding these products. Finally, divide this sum by the total number of units. Ex 3) Find the gradepoint average (weighted mean) for the grades below. Course Grade Points Units (credits) Math 4 (A) 5 History 3 (B) 3 Health 4 (A) 2 Art 2 (C) 2 Cosner  Math Notes  1
2 Median Another measure of central tendency, which is not so sensitive to extreme values, is the median. This measure divides a group of numbers into two parts, with half the numbers below the median and half above it. To find the median of a group of items: 1) Rank the items. (Put in order.) 2) If the number of items is, the median is the middle item in the list. 3) If the number of items is, the median is the mean of the two middle numbers. Ex 4) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 1, 6, 3, 3, 4, 2. Find the median number of siblings for the ten students. Position of the Median in a Frequency Distribution Position of median = Notice that this formula gives the position, and not the actual value. Ex 5) Find the median for the distribution. Value Frequency n 1 f Mode The mode of a data set is the value that occurs most frequently. When two values occur with the same greatest frequency, each one is a mode and the data set is bimodal. When more than two values occur with the same greatest frequency, each ins a mode and the data set is multimodal. When no value is repeated, we say that there is no mode. Ex 6) Ten students in a math class were polled as to the number of siblings in their individual families and the results were: 3, 2, 2, 1, 3, 6, 3, 3, 4, 2. Find the mode for the number of siblings. Cosner  Math Notes  2
3 Ex 7) Find the mode for the distribution. Value Frequency Central Tendency from StemandLeaf Displays We can calculate measures of central tendency from a stemandleaf display. The median and mode are easily identified when the leaves are ranked (in numerical order) on their stems. Ex 8) Given the stem and leaf plot below, calculate the three measures of central tendency. Symmetry in Data Sets The most useful way to analyze a data set often depends on whether the distribution is symmetric or nonsymmetric. In a symmetric distribution, as we move out from a central point, the pattern of frequencies is the same (or nearly so) to the left and right. In a nonsymmetric distribution, the patterns to the left and right are different. A nonsymmetric distribution with a tail extending out to the left, shaped like a J, is called skewed to the left. If the tail extends out to the right, the distribution is skewed to the right. Cosner  Math Notes  3
4 13.3 Measures of Dispersion 13.3 Book HW: 4 (by hand), 12 The mean is a good indicator of the central tendency of a set of data values, but it does not give the whole story about the data. Ex 1) Below are scores on quizzes in 2 classes. Which class did better? Class 1 Class x 6.3 x 6. 3 mode = 6,8 mode = 6 median = 6 median = We cannot tell by just the measures of central tendency. So we must look at the measure of dispersion, or spread, of the data. Two of the most common measures of dispersion are the range and the standard deviation. The range of a set of data is the difference between the maximum value and minimum value. The range is easy to compute; however the range does not take into account all values. So, outliers can affect the range. The standard deviation of a set of sample values is a measure of variation of values about the mean. x x Sample standard deviation: s n 1 The standard deviation is a measure of variation of all values from the mean. Outliers can dramatically increase the standard deviation. The units of the standard deviation, s, are the same as the units of the original data values. The value of the standard deviation is usually positive. Question: When is the standard deviation not positive? 2 Ex 2) Find the mean and standard deviation of the given data (in minutes) by hand and then with calculator: a),,,, x (min) x x x x 2 Totals Cosner  Math Notes  4
5 b) 46,,,, 54 x (min) x x x x Totals c) 5,,,, 95 x (min) x x x x Totals Ex 3) Find the sample standard deviation for the frequency distribution. Value Frequency Cosner  Math Notes  5
6 The sample variance of a set of sample values is a measure of variation equal to the square of the standard deviation. Ex 4) In the preceding example we found that for 46,,,, 54 (minutes), the standard deviation was about 2.8 minutes. Find the variance of that same example. Central tendency and dispersion are different independent aspects of a set of data. Which one is more critical can depend on the situation. For example, suppose muffins sell by the basket. If you want the most muffins possible per dollar spent, you would look for the basket with the highest average weight per muffin. Or, if the muffins are to be served on a tray where presentation is important, you would look for a basket with uniformsized muffins that is a basket with the lowest weight dispersion. So the more desirable the basket depends on your objective. Ex 5) Two companies, A and B, sell 12 ounce jars of instant coffee. Five jars of each were randomly selected from markets, and the contents were carefully weighed, shown below: A B a) Find which company provides more coffee in their jars. b) Find which company fills its jars more consistently. It is clear that a larger dispersion value means more spread than a smaller one. But it is difficult to say exactly what a single dispersion value says about a data set. Cosner  Math Notes  6
7 13.4 Measures of Position 13.4 Book HW: 5, 6, 7, 28, 34 In some cases we are interested in certain individual items in the data set, rather than in the set as a whole. We need a way of measuring how an item fits into the collection, how it compares to other items in the collection, or even how it compares to another item in another collection. There are several common ways of creating such measures and they are usually called measures of position. If x is a data item in a sample with mean x and standard deviation s, then the zscore of x is given by z = A zvalue (or standardized value), is the number of standard deviations that a given value x is above or below the mean. (Round z to two decimal places.) It is found using the following: Ex 1) Two students, who take different history classes, had exams on the same day. Jen s score was 83 while Joy s score was 78. Which student did relatively better, given the class data shown below? Jen Joy Class mean Class standard deviation 4 5 Ex 2) Stanford Binet IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein reportedly had an IQ of 160. a) What is the difference between Einstein s IQ and the mean? b) Convert Einstein s IQ score to a zscore. c) How many standard deviations is that (the difference in part a)? d) If we consider usual IQ scores to be those that covert to z scores between 2 and 2, is Einstein s IQ usual or unusual? Percentiles A percentile is a measure of location which divides the data set into 100 groups with 1% of the values in each group. We denote percentiles as P1, P2, P3,..., P 99. If we talk about P 35, then we mean the 35 th percentile and this means that about 35% of the data values will lie below this value. If we talk about P, then we mean the th percentile and this means that % of the data values lie below this value the median. Additionally, just as when finding the median, when we find percentiles we must order the data set first. Cosner  Math Notes  7
8 There are two ways that we want to talk about percentiles: 1. We may want to know what percentile corresponds to a known data value; or 2. We may want to know what data value corresponds to a particular percentile. To find the percentile corresponding to a known data value we compute number of values less than x percentile of value x = 100 total number of data values (Round to the nearest whole number) Ex 3) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: a) What percentile corresponds to an exam score of 73? b) What percentile corresponds to an exam score of 45? To find the data value corresponding to a stated percentile we must define a few notations: n = total number of values in the data set k = percentile being used L = location of the desired data value in the ORDERED data set th P = k percentile k First, we calculate the value of the location or position, L, as follows: k L n 100 Now, there are two options for L: 1. It is NOT a whole number and then we round UP to the next whole number and locate the data value that occupies that position; OR 2. It is a whole number and then we find the average of the Lth and (L + 1)st numbers in the ordered data set. Cosner  Math Notes  8
9 Ex 4) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: Find the 23 rd percentile for the Chapter 3 exam scores. Find the 70 th percentile for the Chapter 3 exam scores. Quartiles Recall that there are 99 percentiles that divide the data set up into 100 groups. There are 3 quartiles that divide the data set up into 4 groups  Q1, Q2, and Q 3. Quartiles are measures of location, just as percentiles, but each group contains about 25% of the data. How can we connect quartiles back to percentiles? Q P 1 25 Q P median Q 2 P 3 75 So, when we find quartiles, really we aren t finding anything new just thinking back to the corresponding percentile. Ex 5) In a particular class of thirty math students, they took an exam over Chapter 3. The exam scores are as follows: Find the quartiles for the data set. Cosner  Math Notes  9
10 The interquartile range (IQR) is the difference between the upper ( Q 3 ) and lower ( Q 1 ) quartiles. Approximately half the data values fall within the interquartile range. IQR Q3 Q1 The semiinterquartile range is the IQR divided by 2. The midquartile is the sum of the upper and lower quartiles divide by 2. Semiinterquartile = midquartile = Q3 Q1 2 Q3 Q1 2 An outlier is a value that that falls far from what would be considered normal data values. We will define outliers in terms of the interquartile range (IQR). A data value will qualify as an outlier if either of the following conditions are met: o The data value is larger than Q3 1.5 IQR ; or o The data value is smaller than Q1 1.5 IQR. We use a diagram called a boxplot visually display the extreme values (minimum and maximum), the quartiles (lower, median, upper) and the IQR over a number line. We draw a box that shows the IQR with a line through the box at the median. Then we draw in whiskers that extend from the box out to the extreme values. The 5 number summary is just an ordered listing of the important values that are used in the box plot in parentheses: Ex 6) Find the 5 number summary and construct a box plot for the Math exam scores. (minimum, Q 1, Q 2, Q 3, maximum) Deciles are the nine values (denoted D 1, D 2,, D 9 ) along the scale that divide a data set into ten (approximately) equal parts, and quartiles are the three values (Q 1, Q 2, Q 3 ) that divide the data set into four (approximately) equal parts. Ex 7) The following are test scores (out of 100) for a particular math class. Find the sixth decile Cosner  Math Notes  10
11 The Box Plot A box plot, or boxandwhisker plot, involves the median (a measure of central tendency), the range (a measure of dispersion), and the first and third quartiles (measures of position), all incorporated into a simple visual display. For a given set of data, a box plot, or boxandwhisker plot, consists of a rectangular box positioned above a numerical scale, extending from Q 1 to Q 3, with the value of Q 2 (the median) indicated within the box, and with whiskers (line segments) extending to the left and right from the box out to the minimum and maximum data items. Ex 8) Construct a box plot for the weekly study times data shown below Ex 9) Using the data set below to create a boxplot for the data set Minimum: Lower Quartile: Median: Upper Quartile: Maximum: IQR: Cosner  Math Notes  11
12 13.5 The Normal Distribution 13.5 Book HW: 46,48, 52, 54, 60b A random variable that can take on only certain fixed values is called a discrete random variable. A variable whose values are not restricted in this way is a continuous random variable. A normal curve is a symmetric, bellshaped curve. Any random variable whose graph has this characteristic shape is said to have a normal distribution. On a normal curve, if the quantity shown on the horizontal axis is the number of standard deviations from the mean, rather than values of the random variable itself, then we call the curve the standard normal curve. Properties of Normal Curves The graph of a normal curve is bellshaped and symmetric about a vertical line through its center. The mean, median, and mode of a normal curve are all equal and occur at the center of the distribution. Empirical (or ) Rule for Data with a BellShaped Distribution This rule states that for data sets having a distribution that is approximately bellshaped, the following properties apply. About % of all values fall within standard deviation of the mean. About % of all values fall within standard deviation of the mean. About % of all values fall within standard deviation of the mean. Ex 1) Suppose that 280 sociology students take an exam and that the distribution of their scores can be treated as normal. Find the number of scores falling within 2 standard deviations of the mean. Ex 2) IQ scores have a bellshaped distribution with a mean of 100 and variance of 225. What percentage of IQ scores are between 70 and 130? Cosner  Math Notes  12
13 A Table of Standard Normal Curve Areas To answer questions that involve regions other than 1, 2, or 3 standard deviations of the mean we can refer to one of our tools, the calculator. The general syntax of the normalcdf command is: normalcdf( start, end, mean, std_dev) 2 nd VARS (DISTR), 2: normalcdf( The TI83/84 will return the percentage (as a decimal number from 0 to 1) of data points x in a normal distribution with the given mean and standard deviation which lie between x = start and x = end. The standard normal distribution is a normal probability distribution with 1) 0 2) 1 3) Total area under its density curve is equal to 1. Ex 3) Use the calculator to find the percent of all scores that lie between the mean and the following values. a) 1.5 standard deviation above the mean b) 2.62 standard deviations below the mean Ex 4) Find the total area indicated in the region in color below. 0 z z Cosner  Math Notes  13
14 Ex 5) Find the following using the table by first drawing a sketch: a) P(z < 1.58) b) P(z < 1.45) c) P(2.3 < z < 1.45) Interpreting Normal Curve Areas In a standard normal curve, the following three quantities are equivalent. 1) Percentage (of total items that lie in an interval) 2) Probability (of a randomly chosen item lying in an interval) 3) Area (under the normal curve along an interval) Ex 6) The volumes of soda in bottles from a small company are distributed normally with mean 12 ounces and standard deviation.15 ounce. If 1 bottle is randomly selected, what is the probability that it will have more than ounces? Cosner  Math Notes  14
15 Ex 7) The volume of liquid in a bottle of eye drop medicine is normally distributed with a mean of 30.5 ml and standard deviation of 2.5 ml. Calculate the following probabilities of randomly selecting a bottle: a) with less than 32 ml of eye drop medicine. b) with the amount of eye drop medicine between 26.5 and 34 ml. Ex 8) Assuming a normal distribution, find the zvalue meeting the condition that 39% of the area is to the right of z. Ex 9) The shrinkage in length of a certain brand of blue jeans is normally distributed with a mean of 1.35 inches and standard deviation of 0.25 inch. What percent of this brand of jeans will shrink between 1 and 2 inches? Cosner  Math Notes  15
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 information103 Measures of Central Tendency and Variation
103 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 informationDescribing 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 informationChapter 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 informationSTATISTICS 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 informationDescriptive 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 informationA 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 informationWe 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 informationThe 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 information1. 2. 3. 4. Find the mean and median. 5. 1, 2, 87 6. 3, 2, 1, 10. Bellwork 32315 Simplify each expression.
Bellwork 32315 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 informationBrief Review of Median
Session 36 FiveNumber Summary and Box Plots Interpret the information given in the following boxandwhisker plot. The results from a pretest for students for the year 2000 and the year 2010 are illustrated
More informationNumerical 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 informationSTAT 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 informationIntroduction 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 informationx Measures of Central Tendency for Ungrouped Data Chapter 3 Numerical Descriptive Measures Example 31 Example 31: Solution
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
More informationSection 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 informationChapter 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 informationThe Normal Distribution
Chapter 6 The Normal Distribution 6.1 The Normal Distribution 1 6.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Recognize the normal probability distribution
More information! 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 information3: 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 informationCH.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 StemandLeaf Diagrams Median, quartiles, percentiles,
More informationconsider the number of math classes taken by math 150 students. how can we represent the results in one number?
ch 3: numerically summarizing data  center, spread, shape 3.1 measure of central tendency or, give me one number that represents all the data consider the number of math classes taken by math 150 students.
More informationMethods for Describing Data Sets
1 Methods for Describing Data Sets.1 Describing Data Graphically In this section, we will work on organizing data into a special table called a frequency table. First, we will classify the data into categories.
More informationEXAM #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 informationChapter 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 informationAP Statistics Solutions to Packet 2
AP Statistics Solutions to Packet 2 The Normal Distributions Density Curves and the Normal Distribution Standard Normal Calculations HW #9 1, 2, 4, 68 2.1 DENSITY CURVES (a) Sketch a density curve that
More informationHISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
Mathematics Revision Guides Histograms, Cumulative Frequency and Box Plots Page 1 of 25 M.K. HOME TUITION Mathematics Revision Guides Level: GCSE Higher Tier HISTOGRAMS, CUMULATIVE FREQUENCY AND BOX PLOTS
More informationExercise 1.12 (Pg. 2223)
Individuals: The objects that are described by a set of data. They may be people, animals, things, etc. (Also referred to as Cases or Records) Variables: The characteristics recorded about each individual.
More information32 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( ) ( ) 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 informationDescriptive 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 informationExploratory 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 informationDescriptive statistics Statistical inference statistical inference, statistical induction and inferential statistics
Descriptive statistics is the discipline of quantitatively describing the main features of a collection of data. Descriptive statistics are distinguished from inferential statistics (or inductive statistics),
More informationChapter 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 information2.0 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 informationData 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 information8. THE NORMAL DISTRIBUTION
8. THE NORMAL DISTRIBUTION The normal distribution with mean μ and variance σ 2 has the following density function: The normal distribution is sometimes called a Gaussian Distribution, after its inventor,
More informationGCSE 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 informationCh. 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 informationCHAPTER 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 informationDescriptive Statistics
Chapter 2 Descriptive Statistics 2.1 Descriptive Statistics 1 2.1.1 Student Learning Objectives By the end of this chapter, the student should be able to: Display data graphically and interpret graphs:
More informationLecture 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 informationUnit 16 Normal Distributions
Unit 16 Normal Distributions Objectives: To obtain relative frequencies (probabilities) and percentiles with a population having a normal distribution While there are many different types of distributions
More informationFrequency 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 information3.2 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 informationIntroduction to Environmental Statistics. The Big Picture. Populations and Samples. Sample Data. Examples of sample data
A Few Sources for Data Examples Used Introduction to Environmental Statistics Professor Jessica Utts University of California, Irvine jutts@uci.edu 1. Statistical Methods in Water Resources by D.R. Helsel
More informationIntroduction 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 informationChapter 2. The Normal Distribution
Chapter 2 The Normal Distribution Lesson 21 Density Curve Review Graph the data Calculate a numerical summary of the data Describe the shape, center, spread and outliers of the data Histogram with Curve
More informationExploratory Data Analysis
Exploratory Data Analysis Johannes Schauer johannes.schauer@tugraz.at Institute of Statistics Graz University of Technology Steyrergasse 17/IV, 8010 Graz www.statistics.tugraz.at February 12, 2008 Introduction
More informationVariables. Exploratory Data Analysis
Exploratory Data Analysis Exploratory Data Analysis involves both graphical displays of data and numerical summaries of data. A common situation is for a data set to be represented as a matrix. There is
More informationcan be denoted Σx = x + x 2 x 2 ,..., x n is calculated as follows: x = Σx n
Unit 2 Part 1: Measures of Central Tendency A small video recycling business had the following daily sales over a six day period: $305, $285, $240, $376, $198, $264 A single number that is, in some sense
More informationThis is Descriptive Statistics, chapter 2 from the book Beginning Statistics (index.html) (v. 1.0).
This is Descriptive Statistics, chapter from the book Beginning Statistics (index.html) (v..). This book is licensed under a Creative Commons byncsa. (http://creativecommons.org/licenses/byncsa/./)
More informationSTATS8: 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 informationLesson 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 informationEach 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 informationMEI Statistics 1. Exploring data. Section 1: Introduction. Looking at data
MEI Statistics Exploring data Section : Introduction Notes and Examples These notes have subsections on: Looking at data Stemandleaf diagrams Types of data Measures of central tendency Comparison of
More informationMEASURES OF VARIATION
NORMAL DISTRIBTIONS MEASURES OF VARIATION In statistics, it is important to measure the spread of data. A simple way to measure spread is to find the range. But statisticians want to know if the data are
More informationReport 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 informationMathematics Teachers Self Study Guide on the national Curriculum Statement. Book 2 of 2
Mathematics Teachers Self Study Guide on the national Curriculum Statement Book 2 of 2 1 WORKING WITH GROUPED DATA Material written by Meg Dickson and Jackie Scheiber RADMASTE Centre, University of the
More informationMEAN 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 information2. A is a subset of the population. 3. Construct a frequency distribution for the data of the grades of 25 students taking Math 11 last
Math 111 Chapter 12 Practice Test 1. If I wanted to survey 50 Cabrini College students about where they prefer to eat on campus, which would be the most appropriate way to conduct my survey? a. Find 50
More informationSummary of Formulas and Concepts. Descriptive Statistics (Ch. 14)
Summary of Formulas and Concepts Descriptive Statistics (Ch. 14) Definitions Population: The complete set of numerical information on a particular quantity in which an investigator is interested. We assume
More informationGlossary of numeracy terms
Glossary of numeracy terms These terms are used in numeracy. You can use them as part of your preparation for the numeracy professional skills test. You will not be assessed on definitions of terms during
More informationSampling, frequency distribution, graphs, measures of central tendency, measures of dispersion
Statistics Basics Sampling, frequency distribution, graphs, measures of central tendency, measures of dispersion Part 1: Sampling, Frequency Distributions, and Graphs The method of collecting, organizing,
More informationIntroduction to the Practice of Statistics Fifth Edition Moore, McCabe
Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 1.3 Homework Answers 1.80 If you ask a computer to generate "random numbers between 0 and 1, you uniform will get observations
More informationDensity Curve. A density curve is the graph of a continuous probability distribution. It must satisfy the following properties:
Density Curve A density curve is the graph of a continuous probability distribution. It must satisfy the following properties: 1. The total area under the curve must equal 1. 2. Every point on the curve
More informationTHE BINOMIAL DISTRIBUTION & PROBABILITY
REVISION SHEET STATISTICS 1 (MEI) THE BINOMIAL DISTRIBUTION & PROBABILITY The main ideas in this chapter are Probabilities based on selecting or arranging objects Probabilities based on the binomial distribution
More informationChapter 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 informationData 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 informationSTT315 Chapter 4 Random Variables & Probability Distributions KM. Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables
Chapter 4.5, 6, 8 Probability Distributions for Continuous Random Variables Discrete vs. continuous random variables Examples of continuous distributions o Uniform o Exponential o Normal Recall: A random
More informationNumerical 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 informationHomework 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 informationMCQ 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 informationDESCRIPTIVE 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 informationCenter: 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 informationMeans, 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 informationTable 21. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.
Chapter 2. DATA EXPLORATION AND SUMMARIZATION 2.1 Frequency Distributions Commonly, people refer to a population as the number of individuals in a city or county, for example, all the people in California.
More informationDesCartes (Combined) Subject: Mathematics Goal: Statistics and Probability
DesCartes (Combined) Subject: Mathematics Goal: Statistics and Probability RIT Score Range: Below 171 Below 171 Data Analysis and Statistics Solves simple problems based on data from tables* Compares
More informationThe Big 50 Revision Guidelines for S1
The Big 50 Revision Guidelines for S1 If you can understand all of these you ll do very well 1. Know what is meant by a statistical model and the Modelling cycle of continuous refinement 2. Understand
More information6.4 Normal Distribution
Contents 6.4 Normal Distribution....................... 381 6.4.1 Characteristics of the Normal Distribution....... 381 6.4.2 The Standardized Normal Distribution......... 385 6.4.3 Meaning of Areas under
More information3.4 The Normal Distribution
3.4 The Normal Distribution All of the probability distributions we have found so far have been for finite random variables. (We could use rectangles in a histogram.) A probability distribution for a continuous
More informationTYPES OF DATA TYPES OF VARIABLES
TYPES OF DATA Univariate data Examines the distribution features of one variable. Bivariate data Explores the relationship between two variables. Univariate and bivariate analysis will be revised separately.
More informationF. 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 informationDiagrams and Graphs of Statistical Data
Diagrams and Graphs of Statistical Data One of the most effective and interesting alternative way in which a statistical data may be presented is through diagrams and graphs. There are several ways in
More informationDescriptive 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 informationProbability Distributions
Learning Objectives Probability Distributions Section 1: How Can We Summarize Possible Outcomes and Their Probabilities? 1. Random variable 2. Probability distributions for discrete random variables 3.
More information4. 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 informationChapter 5: The normal approximation for data
Chapter 5: The normal approximation for data Context................................................................... 2 Normal curve 3 Normal curve.............................................................
More informationStatistics GCSE Higher Revision Sheet
Statistics GCSE Higher Revision Sheet This document attempts to sum up the contents of the Higher Tier Statistics GCSE. There is one exam, two hours long. A calculator is allowed. It is worth 75% of the
More informationMULTIPLE 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) 12 9 34 22 56
More information= = = pages. = = =399 pages. = = 13 cm. 09 Measures of Center, Spread, and Position. Find the mean, median, and mode for each set of data.
Find the mean, median, and mode for each set of data. 1. number of pages in each novel assigned for summer reading: 224, 272, 374, 478, 960, 394, 404, 308, 480, 624 To find the mean divide the sum of all
More informationMathematics. Probability and Statistics Curriculum Guide. Revised 2010
Mathematics Probability and Statistics Curriculum Guide Revised 2010 This page is intentionally left blank. Introduction The Mathematics Curriculum Guide serves as a guide for teachers when planning instruction
More informationStatistics Summary (prepared by Xuan (Tappy) He)
Statistics Summary (prepared by Xuan (Tappy) He) Statistics is the practice of collecting and analyzing data. The analysis of statistics is important for decision making in events where there are uncertainties.
More informationBNG 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 information2. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of 2006 model motor vehicles.
Math 1530017 Exam 1 February 19, 2009 Name Student Number E There are five possible responses to each of the following multiple choice questions. There is only on BEST answer. Be sure to read all possible
More informationLesson Plan. Mean Count
S.ID.: Central Tendency and Dispersion S.ID.: Central Tendency and Dispersion Summarize, represent, and interpret data on a single count or measurement variable. Use statistics appropriate to the shape
More information2.3. Measures of Central Tendency
2.3 Measures of Central Tendency Mean A measure of central tendency is a value that represents a typical, or central, entry of a data set. The three most commonly used measures of central tendency are
More informationCHINHOYI 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 informationGraphical and Tabular. Summarization of Data OPRE 6301
Graphical and Tabular Summarization of Data OPRE 6301 Introduction and Recap... Descriptive statistics involves arranging, summarizing, and presenting a set of data in such a way that useful information
More information6 3 The Standard Normal Distribution
290 Chapter 6 The Normal Distribution Figure 6 5 Areas Under a Normal Distribution Curve 34.13% 34.13% 2.28% 13.59% 13.59% 2.28% 3 2 1 + 1 + 2 + 3 About 68% About 95% About 99.7% 6 3 The Distribution Since
More information