1. 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. 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.

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

5. Population Mean For ungrouped data (data not in a frequency distribution)

6. A Parameter is a measurable characteristic of a population.
56,000
The Kiers family owns four cars. The following is the current mileage on each of the four cars. The mean mileage for the cars is 40,333 1/3 miles.
42,000
23,000
73,000

7. EXAMPLE – Population Mean

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

9. Sample Mean For ungrouped data (data not in a frequency distribution)

10. EXAMPLE – Sample Mean

11. A statistic is a measurable characteristic of a sample.
A sample of five executives received the following bonus last year ($000):
14.0, 15.0, 17.0, 16.0, 15.0
Find The Sample Mean:

12. 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.

13. Sum Of The Deviations Of Each Value From The Mean Is Zero
Consider the set of values: 3, 8, and 4. The mean is 5.
Later, this will become relevant information when we calculate the variation and standard deviation (This is the reason we will have to square the deviations).

14. 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:

15. Weighted Mean
Instead of adding all the observations, we multiply each observation by the number of times it happens.
The weighted mean of a set of numbers X1, X2, ..., Xn, with corresponding weights w1, w2, ...,wn, is computed from the following formula: or

16. 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?

17. 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 odd set of values, the median will be the middle number.
For an even set of values, the median will be the arithmetic average of the two middle numbers.
Median is the measure of central tendency usually used by real estate agents. Why?...

18. Median Example
2nd example:

19. Median Example in Excel

20. 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.

21. 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

22. Mode
The Mode is the value of the observation that appears most frequently.
The mode is especially useful in describing nominal and ordinal levels of measurement.
There can be more than one mode.

23. Mode Example – Nominal Level Data

24. Mode Example – Nominal Level Data
With Nominal Data you would count to see which occurs most frequently
You can build a Frequency Table using the COUNTIF function in Excel.
The one that occurs the most is the Mode
More data…

25. Example – Mode – Ratio Level Data

26. 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.

27. Mean, Median, Mode Using “Descriptive Statistics” in Excel (From Textbook)

28. The Relative Positions of the Mean, Median and the Mode

29. 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 GM gives a more conservative figure that is not drawn up by large values in the set.
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:

30. Geometric Mean
The GM of a set of n positive numbers is defined as the nth root of the product of n values. The formula is either (both are true):

31. Geometric Mean Example 1: Percentage Increase

32. Verify Geometric Mean Example
The GM gives a more
conservative figure
that is not drawn
up by large values
in the set.

33. 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?

34. Another Use Of GM: Ave. % Increase Over Time
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
Where n = number of periods

35. Example for GM: Ave. % Increase Over Time
The total number of females enrolled in American colleges increased from 755,000 in 1992 to 835,000 in That is, the geometric mean rate of increase is 1.27%.
The annual rate of increase is 1.27%
For the years 1992 through 2000, the rate of female enrollment growth at American colleges was 1.27% per year

36. 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.

37. Dictionary Definitions: Dispersion, Variation, Deviation
The spatial property of being scattered about over an area or volume
The degree of scatter of data, usually about an average value, such as the median or mean
Variation
The act of changing or altering something slightly but noticeably from the norm or standard
Deviation
A variation that deviates from the standard or norm

38. Dispersion How spread out is the data?
What is the average of all the deviations?
A small value for a measure of dispersion indicates that the data are clustered around the typical value (mean)
Mean can fairly represent the data
A large value for a measure of dispersion indicates that the data are not clustered around the typical value (mean)
Mean may not fairly represent the data

39. Samples of Dispersions

40. Measures of Dispersion
Range
Mean Deviation
Variance and Standard Deviation

41. Range Range = Highest Value – Lowest Value Excel = MAX – MIN functions
The difference between the largest and the smallest value
Advantage:
It is easy to compute and understand
Disadvantage:
Only two values are used in its calculation
It is influenced by an extreme value

42. Range Example

43. Mean Deviation (MA or MAD)
Mean Deviation measures the mean amount by which the values in a population, or sample, vary from their mean

44. Variance
In Excel: 1) Population Variance  VARP function ) Sample Variance  VAR function

45. Standard Deviation
In Excel: 1) Population Variance  STDEVP function ) Sample Variance  STDEV function
The primary use of the statistic s2 is to estimate σ2, therefore (n-1) is necessary to get a better representation of the deviation or dispersion in the data (n tends to underestimate)

46. EXAMPLE – Range Range = Largest – Smallest value = 80 – 20 = 60
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

47. 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.

48. 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?
2 ^(1/2) =  =

49. EXAMPLE – Sample Variance and Sample Standard Deviation
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 and sample standard deviation?
s2 ^(1/2) = s =

50. EXAMPLE – Sample Standard Deviation (p1)

51. EXAMPLE – Sample Standard Deviation (p2) Conclusions

52. 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?

53. Chebyshev’s Theorem
For any set of observations (sample or population & does not have to be normal distributed), the proportion of the values that lie within (+/-) k standard deviations of the mean is at least:
where k is any constant greater than 1
2) The smaller this
3) The bigger this
1) The bigger this

54. The Empirical Rule (also called Normal Rule)

57. The Arithmetic Mean of Grouped Data

58. The Arithmetic Mean of Grouped Data - Example
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.

59. The Arithmetic Mean of Grouped Data - Example

60. Standard Deviation of Grouped Data

61. 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

62. Approximate Median from Frequency Distribution
Locate the class in which the median lies
Divide the total number of data values by 2.
Determine which class will contain this value. For example, if n=50, 50/2 = 25, then determine which class will contain the 25th value
Use formula to arrive at estimate of Median:

63. Approximate Median from Frequency Distribution Example 1
40th value must be here
The estimated median vehicle selling price is $19,588.00

64. Let’s think about this:
n/2 – CF = 80/2-31 = 9 vehicles to get from the 31st to the 40th
(n/2 – CF)/f = 9/17 = 9/17th of the distance through the class width of $3000
((n/2 – CF)/f)(i) = (9/17)($3000) = $1,588
((n/2 – CF)/f)(i) + L = $1,588 + $18,000 = $19,588, Median estimate

65. Estimate Range from Grouped Data

66. End of Chapter 3