Elementary Statistics Lecture Notes - Chapter 2.3: Measures of Central Tendency, Summaries of Statistics

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Chapter

Descriptive Statistics

Chapter Outline

• 2.1 Frequency Distributions and Their Graphs
• 2.2 More Graphs and Displays
• 2.3 Measures of Central Tendency
• 2.4 Measures of Variation
• 2.5 Measures of Position

Section 2.3 Objectives

• How to find the mean, median, and mode of a population and of a sample
• How to find the weighted mean of a data set, and how to estimate the sample mean of grouped data
• How to describe the shape of a distribution as symmetric, uniform, or skewed and how to compare the mean and median for each .

Measures of Central Tendency Measure of central tendency

• A value that represents a typical, or central, entry of a data set.
• Most common measures of central tendency:

▪ Mean

▪ Median

▪ Mode

.

Example: Finding a Sample Mean The weights (in pounds) for a sample of adults before starting a weight-loss study are listed. What is the mean weight of the adults? 274 235 223 268 290 285 235 .

Solution: Finding a Sample Mean

• The sum of the weights is Σ x = 274 + 235 + 223 + 268 + 290 + 285 + 235 = 1810
• To find the mean weight, divide the sum of the weights by the number of adults in the sample. The mean weight of the adults is about 258.6 pounds. . 274 235 223 268 290 285 235

Example: Finding the Median Find the median of the weight listed in the first example. 274 235 223 268 290 285 235 .

Solution: Finding the Median

• First, order the data. 223 235 235 268 274 285 290
• There are seven entries (an odd number), the median is the middle, or fourth, data entry. The median weight of the adults is 268 pounds. .

Solution: Finding the Median

• First order the data. 223 235 235 268 274 290
• There are six entries (an even number), the median is the mean of the two middle entries. The median weight of the remaining adults is 251.5 pounds. .

Measure of Central Tendency: Mode

Mode

• The data entry that occurs with the greatest frequency.
• If no entry is repeated the data set has no mode.
• If two entries occur with the same greatest frequency,

each entry is a mode ( bimodal ).

.

Solution: Finding the Mode

• Ordering the data helps to find the mode. 223 235 235 268 274 285 290
• The entry of 235 occurs twice, whereas the other data entries occur only once. The mode of the weights is 235 pounds. .

Example: Finding the Mode At a political debate a sample of audience members was asked to name the political party to which they belong. Their responses are shown in the table. What is the mode of the responses? Political Party Frequency, f Democrat 46 Republican 34 Independent 39 Other/don’t know 5 .

Comparing the Mean, Median, and Mode

• All three measures describe a typical entry of a data set.
• Advantage of using the mean:

▪ The mean is a reliable measure because it takes

into account every entry of a data set.

• Disadvantage of using the mean:

▪ Greatly affected by outliers (a data entry that is far

removed from the other entries in the data set). .

Example: Comparing the Mean, Median, and Mode The table shows the sample ages of students in a class. Find the mean, median, and mode of the ages. Are there any outliers? Which measure of central tendency best describes a typical entry of this data set? Ages in a class 20 20 20 20 20 20 21 21 21 21 22 22 22 23 23 23 23 24 24 65 .