1. Chapter 3 Describing Data Using Numerical Measures
Business Statistics:
A Decision-Making Approach
7th Edition
Chapter 3 Describing Data Using Numerical Measures

2. Chapter Goals After completing this chapter, you should be able to:
Compute and interpret the mean, median, and mode for a set of data
Compute the range, variance, and standard deviation and know what these values mean
Construct and interpret a box and whisker graph
Compute and explain the coefficient of variation and
z scores
Use numerical measures along with graphs, charts, and tables to describe data

3. Chapter Topics Measures of Center and Location Mean, median, mode
Other measures of Location
Weighted mean, percentiles, quartiles
Measures of Variation
Range, interquartile range, variance and standard deviation, coefficient of variation
Using the mean and standard deviation together
Coefficient of variation, z-scores

4. Summary Measures Describing Data Numerically Center and Location
Other Measures of Location
Variation
Mean
Range
Percentiles
Median
Interquartile Range
Quartiles
Mode
Variance
Weighted Mean
Standard Deviation
Coefficient of Variation

5. Measures of Center and Location
Mean
Median
Mode
Weighted Mean

6. Mean (Average) The most common measure of central tendency
(continued)
The most common measure of central tendency
Mean = sum of values divided by the number of values
Affected by extreme values (outliers)
Mean = 3
Mean = 4

7. Median
In an ordered array, the median is the “middle” number, i.e., the number that splits the distribution in half
The median is not affected by extreme values
Useful when data are highly skewed
Median = 3
Median = 3

8. Which measure of location is the “best”?
Mean is generally used, unless extreme values (outliers) exist
Then Median is often used, since the median is not sensitive to extreme values.
Example: Realtors say “Median Home Prices”….not average – less sensitive to outliers

9. Shape of a Distribution based on Mean and Median
Describes how data is distributed
Symmetric or skewed
Symmetric
Right-Skewed
Left-Skewed
Mean < Median
Mean = Median
Median < Mean
(Longer tail extends to left)
(Longer tail extends to right)

10. Mode The value that is repeated most often
Not affected by extreme values
Used for either numerical or categorical data
There may be no mode
There may be several modes
Mode = 5
No Mode

11. Implication Five houses on a hill by the beach
House Prices: $2,000, , , , ,000

12. Implication Mean: ($3,000,000/5) = $600,000
= $600,000
Median: middle value of ranked data = $300,000
Mode: most frequent value = $100,000
House Prices: $2,000,000
500, , , ,000
Sum 3,000,000

13. Practice
From the class website, download “describing numerical data using Excel.”
Try followings:
Mean vs. Median
Hockey
Shoes

14. Weighted Mean
Used when values are grouped by frequency or relative importance (i.e., GPA)
Example: Sample of 26 Repair Projects
Weighted Mean Days to Complete:
Days to Complete
Frequency 5 4 6 12 7 8 2

15. Other Location Measures
Other Measures of Location
Percentiles
Quartiles
A percentile is a measure that tells us what percent of the total frequency scored at or below that measure. 
SAT score 90th percentile: the score is as high as or higher than 90% of other students who took the SAT test.
When statistical data is ordered sequentially, the data can be divided into 4 groups of data.
Each of the groups of data is separated by a line called a Quartile.
Example 3-8 on pp. 99

16. Percentiles
Example 1: Find the 60th percentile in an ordered array of 19 values.
So, use value in the i = 12th position
If i is not an integer, round off to the next higher integer value.
If i is an integer, the pth percentile is the average of the values in position i and i+1.

17. Quartiles Quartiles split the ranked data into 4 equal groups:
Note that the second quartile (the 50th percentile) is the median
25%
25%
25%
25% Q1 Q2 Q3

18. Quartiles 25 Example: Find the first quartile (25th percentile)
Sample Data in Ordered Array:
(n = 9)
Q1 = 25th percentile, so find i : i = (9) = 2.25
so round up and use the value in the 3rd position: Q1 = 13 25
100

19. Practice
Use “describing numerical data using Excel” file, try following:
Salary

20. Box and Whisker Plot
Statistics assumes that your data points (the numbers in your list) are clustered around some central value.
The "box" in the box-and-whisker plot contains and highlights the middle half of data points.
It is useful for quickly finding outliers - data points out of line with the rest of the data set.
The Box and Whisker plot can be shown in either vertical or horizontal format.
Example 3-9 on pp.100

21. Box and Whisker Plot
Outliers are plotted outside the calculated limits and the center box extends from Q1 to Q3.
25% % % % * *
Outliers Lower st Median rd Upper
Limit Quartile Quartile Limit
The lower limit is Q1 – 1.5 (Q3 – Q1)
The upper limit is Q (Q3 – Q1)

22. Box-and-Whisker Plot Example
Below is a Box-and-Whisker plot for the following data:
This data is right skewed, as the plot depicts
Min Q Q Q Max *
Upper limit = Q (Q3 – Q1)
= (6 – 2) = 12
27 is above the upper limit so is shown as an outlier

23. Distribution Shape and Box and Whisker Plot
Left-Skewed
Symmetric
Right-Skewed Q1 Q2 Q3 Q1 Q2 Q3 Q1 Q2 Q3

24. Practice Using Free Online Tool
Using the data set on the website, develop a Box-and-Whisker Plot (free online tool).

25. Coefficient of Variation
Measures of Variation
Variation
Range
Variance
Standard Deviation
Coefficient of Variation
Population
Standard
Deviation
Interquartile
Range
Population
Variance
Sample
Variance
Sample
Standard
Deviation

26. Variation
Measures of variation give information on the spread of the data values.
Same center,
different variation

27. Range (refer to Measures of Variability)
Simplest measure of variation
Difference between the largest and the smallest observations:
Range = xmaximum – xminimum
Example:
Range = = 13

28. Disadvantages of the Range
Ignores the way in which data are distributed
Sensitive to outliers (extreme values)
Range = = 5
Range = = 5
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,5
Range = = 4
1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4,120
Range = = 119

29. Interquartile Range (refer to Measures of Variability)
Eliminate range’s susceptibility to extreme values
Can eliminate some outlier problems
Eliminate some high-and low-valued observations and calculate the range from the remaining values.
Interquartile range = 3rd quartile – 1st quartile
Example 3-10 on pp.108

30. Interquartile Range Example
Median
(Q2) X X Q1 Q3
maximum
minimum
25% % % %
Interquartile range
= 57 – 30 = 27

31. Variance Average of squared deviations of values from the mean
Population variance:
Sample variance:

32. Standard Deviation Most commonly used measure of variation
Shows variation about the mean
Has the same units as the original data
Population standard deviation:
Sample standard deviation:

33. Comparing Standard Deviations
Same mean, but different standard deviations:
Data A
Mean = 15.5
s = 3.338
Data B
Mean = 15.5
s = .9258
Data C
Mean = 15.5
s = 4.57

34. Practice
Try the “Variance Vs. Standard Deviation” link on the class website
Use “describing numerical data using Excel” file, try following:
Std.

35. Coefficient of Variation
The coefficient of variation measures the relative variation for distributions with different means.
In finance, CV measures the relative risk of a stock portfolio
It is usually expressed as a percentage.
Population Sample

36. Comparing Coefficients of Variation
Stock A:
Average price last year = $50
Standard deviation = $5
Stock B:
Average price last year = $100
Both stocks have the same standard deviation, but stock B is less variable relative to its price

37. The Empirical Rule
If the data distribution is bell-shaped, then the interval:
contains about 68% of the values in the population or the sample
68%

38. The Empirical Rule
contains about 95% of the values in the population or the sample
contains about 99.7% of the values in the population or the sample
95%
99.7%

39. Practice: Getting Descriptive Statistics Using Excel
Descriptive Statistics are easy to obtain from Microsoft Excel
Use menu choice: Data / data analysis / descriptive statistics
Enter details in dialog box

40. Open Excel and then enter the data
Select: Data / data analysis / descriptive statistics

41. Using Excel Enter dialog box details Check box for summary statistics
(continued)
Enter dialog box details
Check box for summary statistics
Click OK

42. Excel output Microsoft Excel descriptive statistics output,
using the house price data:
How to read “Skewness”?
House Prices: $2,000,000
500, , , ,000

43. Chapter Summary Described measures of center and location
Mean, median, mode, weighted mean
Discussed percentiles and quartiles
Created Box and Whisker Plots
Illustrated distribution shapes
Symmetric, skewed

44. Chapter Summary Described measure of variation
(continued)
Described measure of variation
Range, interquartile range, variance,
standard deviation, coefficient of variation
Discussed Tchebysheff’s Theorem
Calculated standardized data values