©The McGraw-Hill Companies, Inc. 2008McGraw-Hill/Irwin

Describing Data:Numerical Measures

Chapter 3

2

GOALS

• Calculate the arithmetic mean, weighted mean, median, mode, and geometric mean.• Explain the characteristics, uses, advantages, and disadvantages of each measure of location.• Identify the position of the mean, median, and mode for both symmetric and skewed distributions.• Compute and interpret the range, mean deviation, variance, and standard deviation. • Understand the characteristics, uses, advantages, and disadvantages of each measure of dispersion.• Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations.

3

Parameters v. Statistics

Definition: A parameter is a numerical characteristic of a population.

Example: The fraction of U. S. voters who support Sen. McCain for President is a parameter.

Definition: A statistic is a numerical characteristic of a sample.

Example: If we select a simple random sample of

n = 1067 voters from the population of all U. S. voters, the fraction of people in the sample who support Sen. McCain is a statistic.

4

Characteristics of the Mean

The arithmetic mean is the most widely used measure of location. It requires the interval scale. Its major characteristics are:

– All values are used.– It is unique.– The sum of the deviations from the mean is 0.– It is calculated by summing the values and dividing by the

number of values.

5

Population Mean

For ungrouped data, the population mean is the sum of all the population values divided by the total number of population values:

6

EXAMPLE – Population Mean

7

Sample Mean

For ungrouped data, the sample mean is the sum of all the sample values divided by the number of sample values:

8

EXAMPLE – Sample Mean

9

Properties of the Arithmetic Mean

Every set of interval-level and ratio-level data has a mean. All the values are included in computing the mean. A set of data has a unique mean. The mean is affected by unusually large or small data values. The arithmetic mean is the only measure of central tendency

where the sum of the deviations of each value from the mean is zero.

10

The Mean is Affected by Extreme Values

Suppose that, in the SunCom example, the data value 119 is replaced by an extreme value, 229.

The mean of the original data set was 97.5. The mean of the data set with this new extreme

value is 106.6667. There are many situations in which a data set

naturally has some rather extreme values; e.g., data on personal income – there are relatively few people with extremely large values of personal income. The few extreme values would affect the calculated value of the mean.

11

Weighted Mean

The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula:

Often each weight represents the number of items in the data set having a particular value.

12

EXAMPLE – Weighted Mean

The Carter Construction Company pays its hourly employees $16.50, $19.00, or $25.00 per hour. There are 26 hourly employees, 14 of which are paid at the $16.50 rate, 10 at the $19.00 rate, and 2 at the $25.00 rate. What is the mean hourly rate paid the 26 employees?

13

The Median

The Median is the midpoint of the values after they have been ordered from the smallest to the largest.– There are as many values above the median as below it in

the data array.– For an even set of values, the median will be the arithmetic

average of the two middle numbers.

14

Properties of the Median

There is a unique median for each data set. It is not affected by extremely large or small

values and is therefore a valuable measure of central tendency when such values occur.

It can be computed for ratio-level, interval-level, and ordinal-level data.

It can be computed for an open-ended frequency distribution if the median does not lie in an open-ended class.

15

EXAMPLES - Median

The ages for a sample of five college students are:

21, 25, 19, 20, 22

Arranging the data in ascending order gives:

19, 20, 21, 22, 25.

Thus the median is 21.

The heights of four basketball players, in inches, are:

76, 73, 80, 75

Arranging the data in ascending order gives:

73, 75, 76, 80.

Thus the median is 75.5

16

Example: Using the Median When There are Extreme Values

The U.S. Department of Commerce Bureau of Labor Statistics regularly publishes information about the distribution of personal incomes in the U.S.  This distribution, of course, has a floor value of $0.00, and a relatively few number of extremely large values (Think Bill Gates).  Hence the Bureau uses the median income, rather than the mean, as the appropriate measure of central tendency.

17

The Mode

The mode is the value of the observation that appears most frequently.

18

Example - Mode

19

Mode of Categorical Data

There is one situation in which the mode is the only measure of central tendency that can be used – when we have categorical, or non-numeric data. In this situation, we cannot calculate a mean or a median. The mode is the most typical value of the categorical data.

Example: Suppose I have collected data on religiousaffiliation of citizens of the U.S. The modal, or mostTypical value, is Roman Catholic, since The RomanCatholic Church is the largest religious organization in the U.S.

20

Mean, Median, Mode Using Excel

Table 2–4 in Chapter 2 shows the prices of the 80 vehicles sold last month at Whitner Autoplex in Raytown, Missouri. Determine the mean and the median selling price. The mean and the median selling prices are reported in the following Excel output. There are 80 vehicles in the study. So the calculations with a calculator would be tedious and prone to error.

21

Mean, Median, Mode Using Excel

22

Example: Test Score Data

We have 25 scores on a final exam, as follows:

86, 83, 56, 98, 82, 52, 71, 88, 75, 91, 69, 88, 64, 78,

81, 74, 77, 83, 90, 85, 64, 79, 71, 83, 64.

We want to calculate the mean, median, and mode for

this data, and to look at the relationship among them.

We enter the data in an Excel spreadsheet, click on the

MegaStat pull-down menu, choose Descriptive

Statistics, enter the appropriate range of the input list,

and choose which descriptive statistics we want.

23

The Relative Positions of the Mean, Median and the Mode

24

The Geometric Mean

Useful in finding the average change of percentages, ratios, indexes, or growth rates over time.

It has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the GDP, which compound or build on each other.

The geometric mean will always be less than or equal to the arithmetic mean.

The geometric mean of a set of n positive numbers is defined as the nth root of the product of n values.

The formula for the geometric mean is written:

25

EXAMPLE – Geometric Mean

Suppose you receive a 5 percent increase in salary this year and a 15 percent increase next year. The average annual percent increase is 9.886, not 10.0. Why is this so? We begin by calculating the geometric mean.

098861151051 . ).)(.(GM

26

EXAMPLE – Geometric Mean (2)

The return on investment earned by Atkins construction Company for four successive years was: 30 percent, 20 percent, -40 percent, and 200 percent. What is the geometric mean rate of return on investment?

..).)(.)(.)(.(GM 2941808203602131 44

27

Example: Geometric Mean (3)

The 2006 population size of Duval County was 837,964. The population grew by 7.6% between 2000and 2006. We want to project the size of thepopulation in 2030, assuming that the growth rate remains the same; i.e., 7.6% every 6 years. The Projected population size in 2030 is (1.0764 X 837,964) = 1123245. The average growth rate over the 24 yearsis found by calculating the geometric mean:

The average growth rate is just what we expect.

076.1076.1076.1076.1076.14 GM

28

Dispersion

Why Study Dispersion?– A measure of location, such as the mean or the median,

only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of the data.

– For example, if your nature guide told you that the river ahead averaged 3 feet in depth, would you want to wade across on foot without additional information? Probably not. You would want to know something about the variation in the depth.

– A second reason for studying the dispersion in a set of data is to compare the spread in two or more distributions.

29

Samples of Dispersions

30

Measures of Dispersion

Range

Mean Deviation

Variance and Standard Deviation

31

EXAMPLE – Range

The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the number of cappuccinos sold.

Range = Largest – Smallest value = 80 – 20 = 60

32

EXAMPLE – Mean Deviation

The number of cappuccinos sold at the Starbucks location in the Orange Country Airport between 4 and 7 p.m. for a sample of 5 days last year were 20, 40, 50, 60, and 80. Determine the mean deviation for the number of cappuccinos sold.

33

EXAMPLE – Variance and Standard Deviation

The number of traffic citations issued during the last five months in Beaufort County, South Carolina, is 38, 26, 13, 41, and 22. What is the population variance?

34

EXAMPLE – Sample Variance

The hourly wages for a sample of part-time employees at Home Depot are: $12, $20, $16, $18, and $19. What is the sample variance?

35

Why Variance and Standard Deviation?

Both the variance and the standard deviation givethe same information about the dispersion of the datavalues. Why have both? They are used for differentpurposes. There is a branch of statistics calledAnalysis of Variance, having to do with analyzingcause-and-effect relationships for experimental data.The standard deviation is often a more useful measureof variability, because its value “looks reasonable” as ameasure of variability, and because it has the sameunit of measurement as the data values themselves.

36

Example: Variance and Standard Deviation

The household grocery data set (p. 46, Exercise 30). The range of values is $570 – $41 = $529. The variance is 11,735.98 square dollars (?). The standard deviation is $108.33.

The standard deviation looks like a more “reasonable”

measure of spread of the data, being about one-fifth of

the range.

37

Example: Sample Range, Variance, and Standard Deviation

For the test score data, the range is 98 – 52 = 46.

The variance is 127.79, and the standard deviation is

11.30. All of these values may be found using

MegaStat. Note that the numeric value of the variance

is not readily interpretable in this case, while the

standard deviation makes somewhat more sense

as a measure of the spread of the data values. The

calculated value is about one-fifth of the range of the

data values.

38

Chebyshev’s Theorem

The arithmetic mean biweekly amount contributed by the Dupree Paint employees to the company’s profit-sharing plan is $51.54, and the standard deviation is $7.51. At least what percent of the contributions lie within plus 3.5 standard deviations and minus 3.5 standard deviations of the mean?

39

The Empirical Rule

40

Example: Stanford-Binet IQ Test

The Stanford-Binet IQ test is constructed and scored

so that the score histogram for the entire adult

population is bell-shaped, the population mean is 100,

and the population standard deviation is 16.

What fraction of the adult population have IQ scores

between 84 and 116? Between 52 and 148?

41

The Arithmetic Mean of Grouped Data

We can obtain an approximate sample mean for grouped data if we have the grouped frequency table for the data.

42

Recall in Chapter 2, we constructed a frequency distribution for the vehicle selling prices. The information is repeated below. Determine the arithmetic mean vehicle selling price.

The Arithmetic Mean of Grouped Data - Example

43

The Arithmetic Mean of Grouped Data - Example

44

Standard Deviation of Grouped Data

45

Standard Deviation of Grouped Data - Example

Refer to the frequency distribution for the Whitner Autoplex data used earlier. Compute the standard deviation of the vehicle selling prices

46

End of Chapter 3