1. Chapter 3
Created by Bethany Stubbe and Stephan Kogitz

2. Describing Data: Numerical Measures
Chapter Three
Describing Data: Numerical Measures
GOALS
When you have completed this chapter, you will be able to:
ONE Calculate the arithmetic mean, median, mode, weighted mean, and the geometric mean.
TWO Explain the characteristics, uses, advantages, and disadvantages of each measure of location.
THREE Identify the position of the arithmetic mean, median, and mode for both a symmetrical and a skewed distribution.

3. Describing Data: Numerical Measures
Chapter Three continued
Describing Data: Numerical Measures
GOALS
When you have completed this chapter, you will be able to:
FOUR
Compute and interpret the range, the mean deviation, the variance, and the standard deviation of ungrouped data.
FIVE Explain the characteristics, uses, advantages, and disadvantages of each measure of dispersion.
SIX
Understand Chebyshev’s theorem and the Empirical Rule as they relate to a set of observations.

4. Describing Data: Numerical Measures
Chapter Three continued
Describing Data: Numerical Measures
SEVEN Compute and interpret quartiles, percentiles, and the interquartile range.
EIGHT Compute and understand the coefficient of variation and the coefficient of skewness.

5. Characteristics of the Mean
The arithmetic mean is the most widely used measure of location.
It is calculated by summing the values and dividing by the number of values.

6. Population Mean
The population mean is the sum of all the population values divided by the total number of values:
where µ is the population mean.
N is the total number of observations.
X is a particular value.
 indicates the operation of adding.

7. EXAMPLE 1
The Kiers family owns four cars. The following is the current mileage on each of the four cars:
56,000, 23,000,42,000,73,000
Find the mean.
56,000, 23,000, 42,000, 73,000

8. Sample Mean
The sample mean is the sum of all the sample values divided by the number of sample values:
Where n is the total number of values in the sample.

9. EXAMPLE 2
A sample of five executives received the following bonus last year ($000):
14.0, 15.0, 17.0, 16.0, 15.0

10. Properties of the Arithmetic Mean
Every set of interval- or ratio-level data has a mean.
All the values are included in computing the mean.
A set of data has only one mean. The mean is unique
The mean is a useful measure for comparing two or more populations.
The arithmetic mean is the only measure of central tendency where the sum of the deviations of each value from the mean will always be zero.

11. EXAMPLE 3
Consider the set of values: 3, 8, and 4. The mean is 5. Illustrating the fifth property:

12. 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:

13. EXAMPLE 4
During a one hour period on a hot Saturday afternoon cabana boy Chris served fifty drinks. He sold five drinks for $0.50, fifteen for $0.75, fifteen for $0.90, and fifteen for $ Compute the weighted mean of the price of the drinks.

14. 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.

15. EXAMPLE 5
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.

16. Example 6 The penalty minutes of four hockey players are:
76, 73, 80, 75
Arranging the data in ascending order gives: 73, 75, 76, 80. Thus the median is 75.5, found by (75+76)/2.

17. 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.

18. The Mode
The mode is the value of the observation that appears most frequently.
EXAMPLE 7: The exam scores for ten students are: 81, 93, 84, 75, 68, 87, 81, 75, 81, 87. Because the score of 81 occurs the most often, it is the mode.

19. Geometric Mean
The geometric mean (GM) of a set of n numbers is defined as the nth root of the product of the n numbers. The formula is:
The geometric mean is used to average percents, indexes, and relatives.

20. EXAMPLE 8 The interest rate on three bonds were 5, 41, and 4 percent.
The geometric mean is

21. Example 8 continued The arithmetic mean is (1.05+1.41+1.04)/3 =1.1667
The GM gives a more conservative profit figure because it is not heavily weighted by the rate of 41percent.

22. Geometric Mean continued
Another use of the geometric mean is to determine the percent increase in sales, production or other business or economic series from one time period to another.

23. EXAMPLE 9
The total population of British Columbia increased from 3,470,307 in 1992 to 4,095,934 in That is, the geometric mean rate of increase is %.

24. Range
The range is the difference between the largest and the smallest value.
Only two values are used in its calculation.
It is influenced by an extreme value.
It is easy to compute and understand.

25. Mean Deviation
The Mean Deviation is the arithmetic mean of the absolute values of the deviations from the arithmetic mean.

26. The Mean Deviation The main features of the mean deviation are:
All values are used in the calculation.
It is not unduly influenced by large or small values.
The absolute values are difficult to manipulate.

27. Mean Deviation
The formula is:

28. EXAMPLE 10
The weights of a sample of crates containing books for the bookstore (in kilograms) are:
103, 97, 101, 106, 103
Find the range and the mean deviation.
Range = 106 – 97 = 9

29. Example 11
To find the mean deviation, first find the mean weight.

30. Example 11 continued
The mean deviation is:

31. Population Variance
The population variance is the arithmetic mean of the squared deviations from the population mean.

32. Population Variance The major characteristics are:
All values are used in the calculation.
The units are awkward, the square of the original units.

33. Variance
The formula for the population variance is:

34. Variance
The formula for the sample variance is:

35. EXAMPLE 12 The ages of the Dunn family are: 2, 18, 34, 42
2, 18, 34, 42
What is the population variance?

36. The Population Standard Deviation
The population standard deviation σ is the square root of the population variance.
For EXAMPLE 12, the population standard deviation is 15.19, found by

37. EXAMPLE 13 The hourly wages earned by a sample of five students are:
$7, $5, $11, $8, $6.
Find the variance.

38. Sample Standard Deviation
The sample standard deviation is the square root of the sample variance.
In EXAMPLE 13, the sample standard deviation is 2.30

39. Interpretation and Uses of the Standard Deviation
Chebyshev’s theorem: For any set of observations, the minimum proportion of the values that lie within k standard deviations of the mean is at least:
where k is any constant greater than 1.

40. Interpretation and Uses of the Standard Deviation
Empirical Rule: For any symmetrical, bell-shaped distribution:
About 68% of the observations will lie within 1s the mean,
About 95% of the observations will lie within 2s of the mean
Virtually all the observations will be within 3s of the mean

41. m- 3s
m-2s
m-1s m
m+1s
m+2s
m+ 3s

42. Relative Dispersion
The coefficient of variation is the ratio of the standard deviation to the arithmetic mean, expressed as a percentage:

43. Skewness
Skewness is the measure of the lack of symmetry of the distribution.

44. Skewness The major characteristics of the coefficient of skewness are:
It can range from up to 3.00.
A value of 0 indicates a symmetric distribution.
It is computed as follows:

45. Interquartile Range
The Interquartile range is the distance between the third quartile Q3 and the first quartile Q1.
This distance will include the middle 50 percent of the observations.
Interquartile range = Q3 - Q1

46. EXAMPLE 14
For a set of observations the third quartile is 24 and the first quartile is 10. What is the interquartile range?
The interquartile range is = Fifty percent of the observations will occur between 10 and 24.

47. Box Plots
A box plot is a graphical display, based on quartiles, that helps to picture a set of data.
Five pieces of data are needed to construct a box plot: the Minimum Value, the First Quartile, the Median, the Third Quartile, and the Maximum Value.

48. EXAMPLE 14 continued median min Q1 Q3 max
EXAMPLE 14 continued