X - Xbar : ( 41-50) (48-50) (50-50) (50-50) (54-50) (57-50) Deviations: (note that sum = 0) Squared :

Size: px
Start display at page:

Download "X - Xbar : ( 41-50) (48-50) (50-50) (50-50) (54-50) (57-50) Deviations: (note that sum = 0) Squared :"

Transcription

1 Review Exercises Average and Standard Deviation Chapter 4, FPP, p Dr. McGahagan Problem 1. Basic calculations. Find the mean, median, and SD of the list x = ( ) Mean = (sum x) / 6 = 300 / 6 = 50 Median = 50. Note that the list must be sorted first: ( ) Squared deviations from the mean, using the sorted data. X - Xbar : ( 41-50) (48-50) (50-50) (50-50) (54-50) (57-50) Deviations: (note that sum = 0) Squared : Sum of squared deviations = = = 150 Variance (mean squared deviation) = 150 / 6 = 25 Standard deviation = square root of variance = 5.0 Half a standard deviation is 2.5; those values within a half SD of the mean are in the interval 50 +/- 2.5, that is, from 47.5 to In the list above, this means the values highlighted in red: ( ) A one SD range goes from 45 to 55; this includes all the values except the final 2. ASSIGNED. Problem 2. Which list has the smaller SD? Explain. Information: Both lists have a mean of 50. Don't calculate; explain the reasoning behind your conclusion ASSIGNED. Problem 3. Guessing the mean and the SD of the list X = Part a. Guess whether the average is closer to 1, 5, or 10. Part b. Guess whether the standard deviation is closer to 1, 3, or 6.

2 Problem 4. Relation of mean and median. Part a. For income in the US. The distribution will be right-skewed -- there are a few very large income households, and a much larger number of middle and lower income households. Household income from Census Bureau's Income, Poverty and Health Insurance Coverage in the US, 2007, available from Table A-1, page 31: Median income = $ 50,233 (with standard error reported at 140); Mean income = $67, 609 (SE = 236). The standard error is a measure of the likely error of the estimate, related to the SD of the population and to the size of the sample; we will meet it later in much more detail. Part b. Years of schooling for US citizens over 25 completed may well be left-skewed -- most likely median at 12 to 14 years, with a mean possibly pulled down below this by the few without any high school. Data from Census 2000 for males aged 25 through 34 (19, 902,737 in category): Less than 9 years : 1,077, years : 2,519,649 HS diploma (12) : 5,464,280 Some college (13) : 4,418,322 Associate degree (14): 1,298,577 Bachelor's (16) : 3,773,593 Graduate, prof. (18) : 1,350,824 As an additional exercise, translate this data into percentages and find the mean and median and SD The numbers in parentheses may be taken as the midpoints of the categories for the purposes of this exercise (they are my guesses rather than the Census Bureau's). Weighted mean = (sum (percents * midpoints). So create variables for the midpoints and percents: use (bind midpoints (list )) to create your list of midpoints, and (bind percents (list of your numbers)), then use the EcLS command (sum (* percents midpoints)) = 12.9 years. Note that percents should be a decimal for this to work, and no, you don't have to divide by 7. To find the median, use the command (cusum percents) which will tell you that only the median is in the "some college" category. The distribution is left-skewed, if only slightly. ASSIGNED.Problem 5. Unusual blood pressure? Explain whether or not the given readings would be considered unusual. Hint: standardize the scores by subtracting the mean then dividing by the SD: Problem 6. Sketches of histograms: (i) Left skewed, with highest point at 75 (ii) Symmetric, with highest point at 50 (iii) Right skewed, with highest point at 25 Note that the median will be closer than the mean to the mode or highest point for skewed distributions. The left skewed distribution will have the lowest average, and the right skewed distribution the highest, a. Averages of 40, 50 and 60 are really not in scrambled order. b. Median < average for (iii); median = average for (ii) and median < average for (i). c. SD for histogram (ii) [I know the text says (iii), but SD of (ii) should be easier to judge first] is definitely less than shade the area between the 25 and 75 and you have most of the area shaded in; you should have only about 68 percent of the area shaded in for a "mean +/- one SD" area. Likewise, shading the area between 45 and 55 would give you a narrow central strip, certainly less than 68 %. This leaves 15 as the most likely SD here. d. SD for histogram (iii) will be bigger than for (ii) -- since the mean is off to one side of the highest point, there will be a lot of points to the left of the mean, and this will run up the sum of squared deviations. But note that the SD cannot be as high as 50, for any of the distributions, because this would cover ALL the data.

3 Problem 7. Weights of college students. Men: Average = 66 kg; SD = 9 kg Women: Average = 55 kg; SD = 9 kg (a) Average and SD in pounds. For weights: Men = 2.2 * 66 kg = 145 lbs; women = 2.2 * 55 kg = 121 lbs. For SD: 2.2 * 9 kg. = 19.8 (b) For men, the one SD range will be between 57 and 75 kg; if weights are normally distributed, 68 percent of men should be in this range. (c) Weights of men and women together will probably be BIMODAL; the average of equal numbers of both will be in the middle, at ( ) / 2 = 60.5 kg, and since most men are grouped around 66 kg, and most women around 55 kg, the SD will be much for the combined group than for either separately. Simulation: the RNORM command below generates 5000 random numbers with mean 66 and SD 9 for men: (bind wt0 (rnorm )) and for women: (bind wt1 (rnorm )) Check for whether the mean and sd of these weights agree with what we asked the computer to do. (they won't perfectly agree -- that's what random means!). Use (mean wt0) and (sd wt0) Then create the weights in pounds: (bind lbs0 (* 2.2 wt0)) and (bind lbs1 (* 2.2 wt1)); are means and SDs close to the values in (a)? For part b, (hist wt0) followed by shading the bins from 57 to 75: (shadebins 57 75). For part c, combine the weights of men and women into the variable wts: (bind wts (combine wt0 wt1)), and find mean and SD, and confirm that the SD is larger. The density plot (an outline sketch of a histogram) shows the bimodality more clearly than the histogram: create with the command (density-plot wt0 wt1).

4 ASSIGNED. Problem 8. Average heights of boys and girls. Given: Boys at age 9: 136 cm. at age 11: 146 cm. Average heights of mixed random sample of boys and girls at age 11: 147 cm. Part a. Are boys taller than girls at age 11? Explain your reasoning. Part b. Estimate the average height of boys at age 10. Explain your reasoning. Part c. (not in text). Suppose that the sample of 11 year olds had 600 girls and 400 boys. What would the average height of girls at 11 be? ASSIGNED. Problem 9. Mean, median and outliers. Computer file with 1000 households has incomes in the range $ 5,800 to $ 98,600. By accident, the highest income gets an extra zero, and is recorded as $ 986,000. Part a. Is the average affected? If so, what is the new average? Part b. Is the median affected? If so, what is the new median? ASSIGNED. Problem 10. Law school scores. Incoming students have mean LSAT = 163 and SD = 8. Pick a student at random and guess their score. For each point you are off, you will be penalized (absolute value of actual score - your guess) Part a. What should you guess the score will be? Answer: 163, since the mean will minimize your likely loss. Part b. You have about 1 chance in 3 of being more than 1 SD off. If you are more than 8 points off, you will lose more than $ 8.

5 Problem 11. Root mean square of losses = the standard deviation. Note however that this is not your "average loss" Problem 12." Underclass" Since the percentage of people in poverty has remained roughly constant, can we conclude that there is a "permanent underclass"? Not necessarily, since there is no guarantee that the same individuals are in poverty from one year to the next -- those newly unemployed will join the lowest ranks of the income distribution, and those newly employed will leave it. Note also that the definition of the poverty line has changed over time.

Math 1011 Homework Set 2

Math 1011 Homework Set 2 Math 1011 Homework Set 2 Due February 12, 2014 1. Suppose we have two lists: (i) 1, 3, 5, 7, 9, 11; and (ii) 1001, 1003, 1005, 1007, 1009, 1011. (a) Find the average and standard deviation for each of

More information

Continuing, we get (note that unlike the text suggestion, I end the final interval with 95, not 85.

Continuing, we get (note that unlike the text suggestion, I end the final interval with 95, not 85. Chapter 3 -- Review Exercises Statistics 1040 -- Dr. McGahagan Problem 1. Histogram of male heights. Shaded area shows percentage of men between 66 and 72 inches in height; this translates as "66 inches

More information

MEASURES OF VARIATION

MEASURES 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 information

Expected values, standard errors, Central Limit Theorem. Statistical inference

Expected values, standard errors, Central Limit Theorem. Statistical inference Expected values, standard errors, Central Limit Theorem FPP 16-18 Statistical inference Up to this point we have focused primarily on exploratory statistical analysis We know dive into the realm of statistical

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

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

Density 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: 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 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

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

Models for Discrete Variables

Models for Discrete Variables Probability Models for Discrete Variables Our study of probability begins much as any data analysis does: What is the distribution of the data? Histograms, boxplots, percentiles, means, standard deviations

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

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

CALCULATIONS & STATISTICS

CALCULATIONS & STATISTICS CALCULATIONS & STATISTICS CALCULATION OF SCORES Conversion of 1-5 scale to 0-100 scores When you look at your report, you will notice that the scores are reported on a 0-100 scale, even though respondents

More information

This HW reviews the normal distribution, confidence intervals and the central limit theorem.

This HW reviews the normal distribution, confidence intervals and the central limit theorem. Homework 3 Solution This HW reviews the normal distribution, confidence intervals and the central limit theorem. (1) Suppose that X is a normally distributed random variable where X N(75, 3 2 ) (mean 75

More information

Regression. In this class we will:

Regression. In this class we will: AMS 5 REGRESSION Regression The idea behind the calculation of the coefficient of correlation is that the scatter plot of the data corresponds to a cloud that follows a straight line. This idea can be

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

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

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

RESIDUAL = ACTUAL PREDICTED The following questions refer to this data set:

RESIDUAL = ACTUAL PREDICTED The following questions refer to this data set: REGRESSION DIAGNOSTICS After fitting a regression line it is important to do some diagnostic checks to verify that regression fit was OK One aspect of diagnostic checking is to find the rms error This

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

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

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

Chapter 8 Homework ( ) -- Normal Distribution

Chapter 8 Homework ( ) -- Normal Distribution Chapter 8 Homework (195-198) -- Normal Distribution Dr. McGahagan NOTE: I often abbreviate the text declaration that X is a random variable distributed normally with mean 8 and variance of 144 as " X is

More information

Probability Models for Continuous Random Variables

Probability Models for Continuous Random Variables Density Probability Models for Continuous Random Variables At right you see a histogram of female length of life. (Births and deaths are recorded to the nearest minute. The data are essentially continuous.)

More information

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

Statistics 1040 Dr. Tom McGahagan DATA AND DESCRIPTIVE STATISTICS

Statistics 1040 Dr. Tom McGahagan DATA AND DESCRIPTIVE STATISTICS Statistics 1040 Dr. Tom McGahagan DATA AND DESCRIPTIVE STATISTICS Data --- information on a variable or group of variables, which may be either numeric (examples: income in dollars, weight in pounds) or

More information

Visual Display of Data in Stata

Visual Display of Data in Stata Lab 2 Visual Display of Data in Stata In this lab we will try to understand data not only through numerical summaries, but also through graphical summaries. The data set consists of a number of variables

More information

Mean, Median, Standard Deviation Prof. McGahagan Stat 1040

Mean, Median, Standard Deviation Prof. McGahagan Stat 1040 Mean, Median, Standard Deviation Prof. McGahagan Stat 1040 Mean = arithmetic average, add all the values and divide by the number of values. Median = 50 th percentile; sort the data and choose the middle

More information

Introduction; Descriptive & Univariate Statistics

Introduction; Descriptive & Univariate Statistics Introduction; Descriptive & Univariate Statistics I. KEY COCEPTS A. Population. Definitions:. The entire set of members in a group. EXAMPLES: All U.S. citizens; all otre Dame Students. 2. All values of

More information

Homework 8 Solutions

Homework 8 Solutions Homework 8 Solutions Chapter 5D Review Questions. 6. What is an exponential scale? When is an exponential scale useful? An exponential scale is one in which each unit corresponds to a power of. In general,

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

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

Lab 6: Sampling Distributions and the CLT

Lab 6: Sampling Distributions and the CLT Lab 6: Sampling Distributions and the CLT Objective: The objective of this lab is to give you a hands- on discussion and understanding of sampling distributions and the Central Limit Theorem (CLT), a theorem

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

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

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

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

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 3 (b) 51

MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question. C) (a) 3 (b) 51 Chapter 2- Problems to look at 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) Height (in inches) 1)

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

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

Unit 16 Normal Distributions

Unit 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 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

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

Exploratory Data Analysis. Psychology 3256

Exploratory Data Analysis. Psychology 3256 Exploratory Data Analysis Psychology 3256 1 Introduction If you are going to find out anything about a data set you must first understand the data Basically getting a feel for you numbers Easier to find

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

Nominal Scaling. Measures of Central Tendency, Spread, and Shape. Interval Scaling. Ordinal Scaling

Nominal Scaling. Measures of Central Tendency, Spread, and Shape. Interval Scaling. Ordinal Scaling Nominal Scaling Measures of, Spread, and Shape Dr. J. Kyle Roberts Southern Methodist University Simmons School of Education and Human Development Department of Teaching and Learning The lowest level of

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

Football Player Weight Analysis Computer Lab Canon City High School vs. Pueblo County High School. Mark Heinen September 20, 2014

Football Player Weight Analysis Computer Lab Canon City High School vs. Pueblo County High School. Mark Heinen September 20, 2014 Football Player Weight Analysis Computer Lab Canon City High School vs. Pueblo County High School Mark Heinen September 20, 2014 Table of Contents I. Problem Statement.. Page 3 II. Solution Technique..

More information

Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables 2

Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables 2 Lesson 4 Part 1 Relationships between two numerical variables 1 Correlation Coefficient The correlation coefficient is a summary statistic that describes the linear relationship between two numerical variables

More information

Statistics 100 Binomial and Normal Random Variables

Statistics 100 Binomial and Normal Random Variables Statistics 100 Binomial and Normal Random Variables Three different random variables with common characteristics: 1. Flip a fair coin 10 times. Let X = number of heads out of 10 flips. 2. Poll a random

More information

Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test.

Variables and Data A variable contains data about anything we measure. For example; age or gender of the participants or their score on a test. The Analysis of Research Data The design of any project will determine what sort of statistical tests you should perform on your data and how successful the data analysis will be. For example if you decide

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

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

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

Section 1.3 Exercises (Solutions)

Section 1.3 Exercises (Solutions) Section 1.3 Exercises (s) 1.109, 1.110, 1.111, 1.114*, 1.115, 1.119*, 1.122, 1.125, 1.127*, 1.128*, 1.131*, 1.133*, 1.135*, 1.137*, 1.139*, 1.145*, 1.146-148. 1.109 Sketch some normal curves. (a) Sketch

More information

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS

MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS MATHEMATICS FOR ENGINEERS STATISTICS TUTORIAL 4 PROBABILITY DISTRIBUTIONS CONTENTS Sample Space Accumulative Probability Probability Distributions Binomial Distribution Normal Distribution Poisson Distribution

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

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

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

Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS

Content DESCRIPTIVE STATISTICS. Data & Statistic. Statistics. Example: DATA VS. STATISTIC VS. STATISTICS Content DESCRIPTIVE STATISTICS Dr Najib Majdi bin Yaacob MD, MPH, DrPH (Epidemiology) USM Unit of Biostatistics & Research Methodology School of Medical Sciences Universiti Sains Malaysia. Introduction

More information

M 225 Test 1 A Name (1 point) SHOW YOUR WORK FOR FULL CREDIT!

M 225 Test 1 A Name (1 point) SHOW YOUR WORK FOR FULL CREDIT! M 225 Test 1 A Name (1 point) SHOW YOUR WORK FOR FULL CREDIT! Problem Max. Points Your Points 1-14 14 15 3 16 5 17 4 18 4 19 11 20 9 21 8 22 16 Total 75 1 Multiple choice questions (1 point each) 1. Look

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

STAB22 section 1.1. total = 88(200/100) + 85(200/100) + 77(300/100) + 90(200/100) + 80(100/100) = 176 + 170 + 231 + 180 + 80 = 837,

STAB22 section 1.1. total = 88(200/100) + 85(200/100) + 77(300/100) + 90(200/100) + 80(100/100) = 176 + 170 + 231 + 180 + 80 = 837, STAB22 section 1.1 1.1 Find the student with ID 104, who is in row 5. For this student, Exam1 is 95, Exam2 is 98, and Final is 96, reading along the row. 1.2 This one involves a careful reading of the

More information

Calculation example mean, median, midrange, mode, variance, and standard deviation for raw and grouped data

Calculation example mean, median, midrange, mode, variance, and standard deviation for raw and grouped data Calculation example mean, median, midrange, mode, variance, and standard deviation for raw and grouped data Raw data: 7, 8, 6, 3, 5, 5, 1, 6, 4, 10 Sorted data: 1, 3, 4, 5, 5, 6, 6, 7, 8, 10 Number of

More information

Week 4: Standard Error and Confidence Intervals

Week 4: Standard Error and Confidence Intervals Health Sciences M.Sc. Programme Applied Biostatistics Week 4: Standard Error and Confidence Intervals Sampling Most research data come from subjects we think of as samples drawn from a larger population.

More information

Exercise 1.12 (Pg. 22-23)

Exercise 1.12 (Pg. 22-23) 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 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

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

(i) I count the number of words on a page. On seven pages the word count is 100, 200, 500, 600, 800, 1300, 1600.

(i) I count the number of words on a page. On seven pages the word count is 100, 200, 500, 600, 800, 1300, 1600. STAT301 Solutions 2 (1a) (i) I count the number of words on a page. On seven pages the word count is 100, 200, 500, 600, 800, 1300, 1600. What is the mean and standard deviation for the above data set?

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

DISTRIBUTION FITTING 1

DISTRIBUTION FITTING 1 1 DISTRIBUTION FITTING What Is Distribution Fitting? Distribution fitting is the procedure of selecting a statistical distribution that best fits to a data set generated by some random process. In other

More information

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement

Research Variables. Measurement. Scales of Measurement. Chapter 4: Data & the Nature of Measurement Chapter 4: Data & the Nature of Graziano, Raulin. Research Methods, a Process of Inquiry Presented by Dustin Adams Research Variables Variable Any characteristic that can take more than one form or value.

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

AP Statistics Solutions to Packet 2

AP 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, 6-8 2.1 DENSITY CURVES (a) Sketch a density curve that

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

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

Estimating and Finding Confidence Intervals

Estimating and Finding Confidence Intervals . Activity 7 Estimating and Finding Confidence Intervals Topic 33 (40) Estimating A Normal Population Mean μ (σ Known) A random sample of size 10 from a population of heights that has a normal distribution

More information

4. Continuous Random Variables, the Pareto and Normal Distributions

4. Continuous Random Variables, the Pareto and Normal Distributions 4. Continuous Random Variables, the Pareto and Normal Distributions A continuous random variable X can take any value in a given range (e.g. height, weight, age). The distribution of a continuous random

More information

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Introduction to the Practice of Statistics Fifth Edition Moore, McCabe Section 5.2 Homework Answers 5.29 An automatic grinding machine in an auto parts plant prepares axles with a target diameter µ = 40.125

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

MEASURES OF DISPERSION

MEASURES OF DISPERSION MEASURES OF DISPERSION Measures of Dispersion While measures of central tendency indicate what value of a variable is (in one sense or other) average or central or typical in a set of data, measures of

More information

Univariate Descriptive Statistics

Univariate Descriptive Statistics Univariate Descriptive Statistics Displays: pie charts, bar graphs, box plots, histograms, density estimates, dot plots, stemleaf plots, tables, lists. Example: sea urchin sizes Boxplot Histogram Urchin

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

Answers Investigation 4

Answers Investigation 4 Applications 1. a. Median height is 15.7 cm. Order the 1 heights from shortest to tallest. Since 1 is even, average the two middle numbers, 15.6 cm and 15.8 cm. b. Median stride distance is 124.8 cm. Order

More information

Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade

Answer: C. The strength of a correlation does not change if units change by a linear transformation such as: Fahrenheit = 32 + (5/9) * Centigrade Statistics Quiz Correlation and Regression -- ANSWERS 1. Temperature and air pollution are known to be correlated. We collect data from two laboratories, in Boston and Montreal. Boston makes their measurements

More information

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo

Describe what is meant by a placebo Contrast the double-blind procedure with the single-blind procedure Review the structure for organizing a memo Readings: Ha and Ha Textbook - Chapters 1 8 Appendix D & E (online) Plous - Chapters 10, 11, 12 and 14 Chapter 10: The Representativeness Heuristic Chapter 11: The Availability Heuristic Chapter 12: Probability

More information

Introduction to Statistics for Computer Science Projects

Introduction to Statistics for Computer Science Projects Introduction Introduction to Statistics for Computer Science Projects Peter Coxhead Whole modules are devoted to statistics and related topics in many degree programmes, so in this short session all I

More information

The correlation coefficient

The correlation coefficient The correlation coefficient Clinical Biostatistics The correlation coefficient Martin Bland Correlation coefficients are used to measure the of the relationship or association between two quantitative

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

Introduction to the Practice of Statistics Fifth Edition Moore, McCabe

Introduction 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 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

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

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

2.3. Measures of Central Tendency

2.3. 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 information

Table 2-1. Sucrose concentration (% fresh wt.) of 100 sugar beet roots. Beet No. % Sucrose. Beet No.

Table 2-1. 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 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

Frequency Distributions

Frequency Distributions Descriptive Statistics Dr. Tom Pierce Department of Psychology Radford University Descriptive statistics comprise a collection of techniques for better understanding what the people in a group look like

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

2. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of 2006 model motor vehicles.

2. Here is a small part of a data set that describes the fuel economy (in miles per gallon) of 2006 model motor vehicles. Math 1530-017 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 information

Use Measures of Central Tendency and Dispersion. Measures of Central Tendency

Use Measures of Central Tendency and Dispersion. Measures of Central Tendency 13.6 Use Measures of Central Tendency and Dispersion Before You analyzed surveys and samples. Now You will compare measures of central tendency and dispersion. Why? So you can analyze and compare data,

More information

Chapter 4: Average and standard deviation

Chapter 4: Average and standard deviation Chapter 4: Average and standard deviation Context................................................................... 2 Average vs. median 3 Average.................................................................

More information