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

Objectives Introduction to Statistical Measures Measures of Central Tendency: Mean, Median, Mode Measures of Spread: Variance and Standard Deviation

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. 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.

What do each of these tell us There are 3 Measures of Central Tendency: Mean, Median, Mode What do each of these tell us about the Data?

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. Which value is a more “typical” value? Why?

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.

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.

Measures of Central Tendency: Mean, Median, Mode 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.

Organizing Data: Frequency Tables Sometimes listing the data in a special way can help us get useful information about the data. One such method the frequency table. 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.

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

Example 4 – Solution The mode is 5, because more students got this score than any other score. The total number of quizzes is 16 + 8 + 5 + 5 + 3 + 3 = 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

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.

Measures of Spread: 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, find the mean of each of the following sets of numbers. 50, 58, 78, 81, 93 72, 71, 72, 72, 73 Although, the means are the same, we say that the first data set shows more variability than the second.

Measures of Spread: Standard Deviation The most important measure of variability in statistics is the standard deviation. Standard deviation measures the average deviation (or difference) from the mean.

Example: Finding the Sample Standard Deviation The starting salaries are for the Chicago branches of a corporation. The corporation has several other branches, and you plan to use the starting salaries of the Chicago branches to estimate the starting salaries for the larger population. Find the sample standard deviation of the starting salaries. Starting salaries (1000s of dollars) 41 38 39 45 47 41 44 41 37 42 Larson/Farber 4th ed. 16

STANDARD DEVIATION The standard deviation, s, is measured in the same units as the original data using the formula:

Solution: Finding the Sample Standard Deviation The sample standard deviation is about 3.1, or $3100. Larson/Farber 4th ed. 18

Solution: Using Technology to Find the Standard Deviation Sample Mean Sample Standard Deviation Larson/Farber 4th ed. 19

Example: Using Technology to Find the Standard Deviation Sample office rental rates (in dollars per square foot per year) for Miami’s central business district are shown in the table. Use a calculator to find the mean rental rate and the sample standard deviation. (Adapted from: Cushman & Wakefield Inc.) Office Rental Rates 35.00 33.50 37.00 23.75 26.50 31.25 36.50 40.00 32.00 39.25 37.50 34.75 37.75 37.25 36.75 27.00 35.75 26.00 29.00 40.50 24.50 33.00 38.00 Larson/Farber 4th ed. 20

Interpreting Standard Deviation Standard deviation is a measure of the typical amount an entry deviates from the mean. The more the entries are spread out, the greater the standard deviation. Larson/Farber 4th ed. 21

Thinking About Variation Since Statistics is about variation, spread is an important fundamental concept of Statistics. Measures of spread help us talk about what we don’t know. When the data values are tightly clustered around the center of the distribution, standard deviation will be small. When the data values are scattered far from the center, standard deviation will be large.

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?

Example 7 – Solution cont’d The standard deviations x and 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.