Probability and Statistics. Copyright Cengage Learning. All rights reserved.

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1 Probability and Statistics Copyright Cengage Learning. All rights reserved.

2 14.5 Descriptive Statistics (Numerical) Copyright Cengage Learning. All rights reserved.

3 Objectives Measures of Central Tendency: Mean, Median, Mode Organizing Data: Frequency Tables and Stemplots Measures of Spread: Variance and Standard Deviation The Five-Number Summary: Box Plots 3

4 Descriptive Statistics (Numerical) Our world is filled with data: weather data, economic data, health data, medical data, political data, census data, and many more. The things that a data set describes are called individuals. So individuals could be people, trees, buildings, etc. The property of the individuals that is described by the data is called a variable. In this chapter we study one-variable data, in which only one property of each individual is measured. 4

5 Descriptive Statistics (Numerical) Data usually consist of thousands or even millions of numbers. The first goal of statistics is to describe such huge sets of data in simpler terms. A summary statistic is a single number that summarizes some property of the data. For example, the average of a data set is a single number that tells us where the center of the data lies. In this section we study several summary statistics. 5

6 Measures of Central Tendency: Mean, Median, Mode 6

7 Measures of Central Tendency: Mean, Median, Mode One way to make sense of data is to find a typical number or the center of the data. Any such number is called a measure of central tendency. One measure of central tendency is the average (or the mean). 7

8 Measures of Central Tendency: Mean, Median, Mode Another measure of central tendency is the median, which is the middle number of an ordered list of numbers there are as many data points above the median as there are below it. 8

9 Measures of Central Tendency: Mean, Median, Mode If a data set includes a number that is far out or far away from the rest of the data, that data point is called an outlier. In general, when a data set has outliers, the median is a better indicator of central tendency than the mean. 9

10 Example 3 Effect of Outliers on the Mean and Median The following table gives the selling prices of houses sold in 2007 in a small coastal California town. (a) Find the mean house price. (b) Find the median house price. (c) The house on Ocean Avenue is resold the same year for $2,400,000. Find the mean and median house price for the adjusted data. 10

11 Example 3 Effect of Outliers on the Mean and Median (d) Comment on the effects of outliers on the mean and median. cont d Solution: (a) Writing the prices in thousands of dollars, we have So the average house price is $500,

12 Example 3 Solution cont d (b) To find the median, we first arrange the prices in order Since there is an even number of prices, the median is the average of the middle two numbers: So the median house price is $250,

13 Example 3 Solution cont d (c) For the revised data So the mean price for the revised data is $600,000. But the median price is unchanged at $250,

14 Example 3 Solution cont d (d) The $1,800,000 price is an outlier. Because of this outlier the average price of $500,000 is not a typical price (the majority of houses sold for prices in the 200,000s). The median of $250,000 is a more typical price. We observe that changing the value of an outlier affects the average but not the median (as long as the change in the outlier keeps the outlier on the same side of the median). 14

15 Measures of Central Tendency: Mean, Median, Mode The mode of a data set is a summary statistic that is usually less informative than the mean or median, but has the advantage of not being limited to numerical data. 15

16 Measures of Central Tendency: Mean, Median, Mode For example, the mode of the (non-numerical) data consisting of Korean last names is Kim. The mode of the data set 1, 1, 2, 2, 2, 3, 5, 8 is the number 2. The data set 1, 2, 2, 3, 5, 5, 8 has two modes: 2 and 5. Data sets with two modes are called bimodal. The data set 1, 2, 4, 5, 7, 8 has no mode. 16

17 Organizing Data: Frequency Tables and Stemplots 17

18 Organizing Data: Frequency Tables and Stemplots Sometimes listing the data in a special way can help us get useful information about the data. Two such methods are frequency tables and stemplots. A frequency table for a set of data is a table that includes each different data point and the number of times that point occurs in the data. The mode is most easily determined from a frequency table. 18

19 Example 4 Using a Frequency Table The scores obtained by the students in an algebra class on a five-question quiz are given in the following frequency table. Find the mean, median, and mode of the scores. Frequency Table 19

20 Example 4 Solution The mode is 5, because more students got this score than any other score. The total number of quizzes is = 40. To find the mean, we add all the scores and divide by 40. Note that the score 5 occurs 16 times, the score 4 occurs 8 times, and so on. So the mean score is 20

21 Example 4 Solution cont d There are 40 students in this class. If we rank the scores from highest to lowest, the median score is the average of the 20th and 21st scores. From frequency column in the table, we see that these scores are each 4. So the median score is 4. 21

22 Organizing Data: Frequency Tables and Stemplots A stemplot (or a stem-and-leaf plot) organizes the data by using the digits of the numbers in the data. Each number in the data is written as a stem consisting of the leftmost digits and a leaf consisting of the rightmost digit. Numbers with the same stem are grouped together in a row, with the stem written only once. The next example illustrates the method. 22

23 Example 5 Making a Stem-and-Leaf Plot A county swim club records the times (in seconds) shown below for the 50-yard freestyle swim event. (a) Make a stem-and-leaf plot of the data. (b) Find the median and mode using the stemplot from part (a). 23

24 Example 5(a) Solution We use the first two digits for the stems and the digit after the decimal point for the leaves. For example, 39.5 has stem 39 and leaf 5. 24

25 Example 5(b) Solution cont d There are 16 data points. The stem-and-leaf plot displays data in increasing order. So to find the median, we count the leaves starting at the smallest data point until we reach the middle two numbers (the eighth and ninth leaves). The corresponding data points are 42.7 and 42.8, so the median is The mode is the number that appears most often in the data. We can see from the stem-and-leaf plot that this number is

26 Measures of Spread: Variance and Standard Deviation 26

27 Measures of Spread: Variance and Standard Deviation Measures of central tendency identify the center or typical value of the data. Measures of spread (also called measures of dispersion) describe the spread or variability of the data around a central value. For example, each of the following data sets has mean 72, but it is clear that the first set of data has more variability than the second. 50, 58, 78, 81, 93 72, 71, 72, 72, 73 27

28 Measures of Spread: Variance and Standard Deviation The most important measures of variability in statistics are the variance and standard deviation. 28

29 Example 7 Calculating Standard Deviation Two machines are used in filling 16-ounce soda bottles. To test how consistently each machine fills the bottles, a sample of 20 bottles from the output of each machine is selected. Find the standard deviation for each machine. Which machine is more consistent in filling the bottles? 29

30 Example 7 Solution The first step is to find the mean for each soda machine: 30

31 Example 7 Solution cont d The standard deviations σ x and σ y for Soda Machines I and II, respectively, are Soda Machine I is more consistent in filling the bottles because the standard deviation of the data from Machine I is much smaller than that of the data from Machine II. 31

32 The Five-Number Summary: Box Plots 32

33 The Five-Number Summary: Box Plots A simple indicator of the spread of data is the location of the minimum and maximum values. Other indicators of spread are the quartiles. As we know that the median divides a data set in half. The median of the lower half of the data is called the first quartile, Q 1. The median of the upper half of the data is called the third quartile, Q 3. 33

34 The Five-Number Summary: Box Plots The minimum, maximum, median, and first and third quartiles together give a good picture of the spread and center of the data. 34

35 The Five-Number Summary: Box Plots Figure 3 shows how these five numbers are related for a data set listed in order. For the data in the figure, Q 1 = (2 + 4)/2 = 3 and Q 3 = ( )/2 = 18. The five-number summary Figure 3 35

36 The Five-Number Summary: Box Plots The five-number summary allows us to quickly calculate the following two different indicators of spread. A simple indicator of spread is the range, which is the difference between the maximum and minimum values: Range = Maximum Minimum This can be compared to the spread of the middle of the data as measured by the interquartile range (IQR), which is the difference between the third and first quartiles: IQR = Q 3 Q 1 36

37 The Five-Number Summary: Box Plots For the data in Figure 3 the range is 41 1 = 40, and the interquartile range is IQR = 18 3 = 15. The five-number summary Figure 3 37

38 Example 8 Using a Five-Number Summary The lists in the margin give HDL cholesterol levels ( good cholesterol ) for 17 male and 13 females who participated in a health survey. Find the five-number summary for each group. Compare the center and the spread in cholesterol levels for the two groups. Solution: For each group we first arrange the data in ascending order and then find the median. Next we find Q 1, the median of the lower half of the data, and Q 3, the median of the upper half of the data. 38

39 Example 8 Solution cont d You can check that the five-number summaries are as follows: Males: 29, 30.5, 41, 48, 55 Females: 46, 49.5, 53, 55.5, 60 The range and interquartile range are calculated as follows: Males: Range = = 26 IQR = = 17.5 Females: Range = = 14 IQR = =

40 Example 8 Solution cont d We first observe that the median HDL level for females, 53, is significantly higher than that of the males, 41. Moreover, female HDL levels show considerably less variability than the males, as measured by the IQR. 40

41 The Five-Number Summary: Box Plots A box plot (also called a box-and-whisker plot) is a method for graphically displaying a five-number summary. The plot consists of a rectangle whose left- and right-hand sides correspond to Q 1 and Q 3, respectively. The box is divided by a line segment at the location of the median. The whiskers are line segments that extend from both sides of the box to the location of the maximum and minimum values. 41

42 The Five-Number Summary: Box Plots Figure 5 shows a box plot for the data in Figure 3. Figure 5 42

43 The Five-Number Summary: Box Plots When we are working with quartiles, the median of the data is also called the second quartile, Q 2. So the three quartiles Q 1, Q 2 and Q 3, separate the data into four parts. Statisticians also use quintiles, deciles, and percentiles. The familiar percentiles are numbers that divide the data into 100 parts, each with the same number of data points. Standardized test scores are often reported as percentiles. So a score in the 99th percentile is a score in the top one-hundredth part of the scores on that test. 43