Section 2.3: One Quantitative Variable: Measures of Spread Objectives: 1) Measures of spread, variability a. Range b. Standard deviation i. Formula ii. Notation for samples and population 2) The 95% rule (Empirical Rule) for bell shaped distributions a. Usual and unusual values 3) Z-scores a. Usual and unusual values 4) Percentiles; quartiles 5) Box plots a. The five number summary b. Range c. Interquartile range d. Which is resistant to outliers? The range or the IQR? 6) Choosing measures of the center and spread a. Mean and median versus five number summary 1
Section 2.3: One Quantitative Variable: Measures of Spread To do at home (pages 1 and 2 only): Read section 2.3, starting on page 77; then, answer the questions. Using the Calculator to Compute Summary Statistics 1) In example 2.15, page 78, Des Moines Versus San Francisco Temperatures a. The variable is: b. We will calculate the mean and median of the data for the two cities. Notice that the answers are in the book. However, you need to practice with the calculator; try it, here are the instructions. FIRST ENTER THE DATA c. Enter the data for Des Moines in List 5 of your calculator. (Press STAT, select Edit) d. Enter the data for San Francisco in List 6 of your calculator. (Press STAT, select Edit) e. PRESS 2 nd MODE[QUIT] to get out of the editor SECOND, CALCULATE THE MEAN AND MEDIAN of each data set f. Press STAT, arrow right to CALC, select 1:VarStats and indicate List 5, press ENTER Note 1: to select L5 do 2 nd number 5 key Note 2: In an Older calculators it will look like: 1-VarStats L5 Record the mean and the median in the table shown below g. Now do the same for list 6 Press STAT, arrow right to CALC, select 1:VarStats and indicate List 6, press ENTER Note 1: to select L6 do 2 nd number 6 key Note 2: In the Older calculators it will look like 1-VarStats L6 Record the mean and the median in the table shown below City Mean Median Des Moines San Francisco h. What do you notice about the measures of the center (mean and median)? Are they very different or almost equal? i. Now explore the dotplots of the data which are shown here. Are these sets equal? What difference do you notice? 2
Section 2.3: One Quantitative Variable: Measures of Spread Standard Deviation 2) Definition of Standard Deviation read the book all answers are there a) What does the standard deviation measure? b) Write the formula to find the standard deviation of a sample of n numbers. c) Answer the following: i. TRUE/FALSE: If false, write the correct statement. The standard deviation gives a rough estimate of the typical distance of a data value from the mean. ii. TRUE/FALSE: If false, write the correct statement. The larger the standard deviation the less variability there is in the data. iii. TRUE/FALSE: If false, write the correct statement. The larger the standard deviation the more spread out are the data. 3) Back to the example about the temperatures in Des Moines and San Francisco. Look at the dotplots for the data shown on the prior page; which distribution do you think will have a larger standard deviation? 4) Let s run again a 1-Var Stats into each of the lists L5 and L6 and record the standard deviation s for each of the two cities. City Des Moines San Francisco Standard Deviation 5) Complete the table with the proper notation for the mean and the standard deviation Notation for the mean Notation for the standard deviation sample population 3
Section 2.3: One Quantitative Variable: Interpreting the Standard Deviation The 95% rule Empirical Rule This rule applies to distributions with a shape We can say that about 95% of the data falls within Common, usual values are within standard deviations from the mean. Unusual values are more than standard deviations from the mean. Figure 2.19 Most data are within two standard deviations of the mean 6) Percent of Body Fat in Men The variable BodyFat in the BodyFat dataset gives the percent of weight made up of body fat for 100 men. For this sample, the mean percent body fat is 18.6 and the standard deviation is 8.0. The distribution of the body fat values is roughly symmetric and bell-shaped. Find an interval that is likely to contain roughly 95% of the data values. About 95% of body fat values are between and Make up your own values to fill in the blanks: It s usual for the percent body fat to be % ; while % is an unusually high value. 4
Section 2.3: One Quantitative Variable: 95% Rule 7) Read example 2.17 on the book page 80 Pulse Rate from Student survey. Complete the following: a. What is the variable? b. What is the shape of the distribution? c. What is the mean x-bar? d. What is the standard deviation s? e. Show the work to identify the rates that are within two standard deviations from the mean. f. Sketch the distribution of pulse rates labeling one, two and three standard deviations around the mean. g. Roughly 95% of are between and h. An example of an unusually low pulse rate is beats per minute. i. 105 beats per minute is an pulse rate. 5
Section 2.3: One Quantitative Variable: Z-scores Z-Scores: The z-score is the number of standard deviations a value is from the mean. Write the formula to find the z-score z-scores for usual values: z scores for unusual values: 8) Percent of Body Fat in Men continued this is the same topic as problem 6, previous page For the sample of 100 men, the mean percent body fat is 18.6 and the standard deviation is 8.0. The largest percent body fat of any man in the sample is 40.1 and the smallest is 3.7. Find and interpret the z-score for each of these values. Which is relatively more extreme? Which is usual? Which is unusual? 9) Read Example 2.19, page 82 of our book - For the patient described in the problem, (ID#772) in the ICU study. He had a high systolic blood pressure of mmhg and a low pulse rate of bpm. The summary statistics for systolic blood pressure show a mean of 132.2 and standard deviation of 32.95, while the heart rates have a mean of 98.9 and standard deviation of 26.83. Which of these values is more unusual relative to the other patients in the sample? 6
Section 2.3: One Quantitative Variable: Measures of Spread 10) Percent Obese by State - Computer output giving descriptive statistics for the percent of the population that is obese for each of the 50 US states, from the USStates dataset, is given in Figure 2.27. Since all 50 US states are included, this is a population, not a sample. (a) What are the mean and the standard deviation? Include appropriate notation with your answers. (b) Calculate the z-score for the largest value and interpret it in terms of standard deviations. Do the same for the smallest value. Are they usual or unusual values? Which one is more extreme? (c) This distribution is relatively symmetric and bell-shaped. Give an interval that is likely to contain about 95% of the data values. (d) Let s introduce the BOX PLOT which is discussed in detail in section 2.4. Sketch a box plot and interpret _ 21 23 25 27 29 31 33 35 7
Section 2.3: One Quantitative Variable Percentiles Percentiles The P th percentile is the value of a quantitative variable which is greater than P percent of the data. 11) Example: John scored in the 90 th percentile in the Math SAT; which of the following is the correct meaning? There are two correct choices. a. 90% of the students who took the same test scored more than or equal to John. b. 90% of the students who took the same test scored less than or equal to John. c. John got 90 points correct out of 100 possible points. d. John s score is greater than or equal to 90% of the scores. 12) Example: Percentiles of SAT Scores A score of 400 on the SAT Mathematics General Test is at the 16 th percentile for all 2012 college-bound seniors taking the SAT. Clearly explain in terms of SAT scores what it means to be at the 16 th percentile. Five Number Summary What values make up the five number summary? Box Plot 13) Which percentile is referred as: a) The first quartile Q1 b) The median Q2 c) The third quartile Q3 Range and Interquartile range How do you find the range? Is it affected by outliers? How do you find the interquartile range? Is it affected by outliers? 8
Section 2.3: One Quantitative Variable Choosing Measures of the Center and Spread Mean and standard deviation versus Five Number Summary Because the standard deviation measures how much the data values deviate from the mean, it makes sense to use the standard deviation as a measure of variability when the mean is used as a measure of center. Advantages: both use all the data values in their calculation Disadvantages: they are not resistant to outliers. The median and IQR are resistant to outliers. Furthermore, if there are outliers or the data are heavily skewed, the five number summary can give more information (such as direction of skewness) than the mean and standard deviation. 14) A Dotplot for the data of Arsenic Concentrations in Toenails is shown Figure 2.29 - Dotplot of arsenic concentration in toenails (a) Which measures of center and spread are most appropriate for this distribution: the mean and standard deviation or the five number summary? Explain. (b) Is it appropriate to use the general rule about having 95% of the data within two standard deviations for this distribution? Why or why not? 15) Which measures of center and spread are most appropriate for this distribution: the mean and standard deviation or the five number summary? Explain. 9