1. Chapter 5
Describing Distributions Numerically

2. Describing the Distribution
Center
Median (.5 quartile, 2nd quartile, 50th percentile)
Mean
Spread
Range
Interquartile Range
Standard Deviation

3. Median Literally = middle number (data value)
Has the same units as the data
n (number of observations) is odd
Order the data from smallest to largest
Median is the middle number on the list
(n+1)/2 number from the smallest value
Ex: If n=11, median is the (11+1)/2 = 6th number from the smallest value
Ex: If n=37, median is the (37+1)/2 = 19th number from the smallest value

4. Example – Frank Thomas 15 observations Median = 32 HRs
Career Home Runs
Remember to order the values, if they aren’t already in order!
15 observations
(15+1)/2 = 8th observation from bottom
Median = 32 HRs

5. Median n is even Order the data from smallest to largest
Median is the average of the two middle numbers
(n+1)/2 will be halfway between these two numbers
Ex: If n=10, (10+1)/2 = 5.5, median is average of 5th and 6th numbers from smallest value

6. Example – Ryne Sandberg
16 observations
(16 + 1)/2 = 8.5, average of 8th and 9th observations from bottom
Median = average of 16 and 19
Median = 17.5 HRs
Career Home Runs
Remember to order the values if they aren’t already in order!

7. Mean Ordinary average Formula Add up all observations
Divide by the number of observations
Has the same units as the data
Formula
n observations
y1, y2, y3, …, yn are the values

8. Mean

9. Examples
Thomas
Sandberg
FIND THE MEAN

10. Mean vs. Median Median = middle number
Mean = value where histogram balances
Mean and Median similar when
Data are symmetric
Mean and median different when
Data are skewed
There are outliers

11. Mean vs. Median
Mean influenced by unusually high or unusually low values
Example: Income in a small town of 6 people
$25,000 $27,000 $29,000
$35,000 $37,000 $38,000
**The mean income is $31,830
**The median income is $32,000

12. Mean vs. Median Bill Gates moves to town Mean is pulled by the outlier
$25,000 $27,000 $29,000
$35,000 $37,000 $38,000 $40,000,000
**The mean income is $5,741,571
**The median income is $35,000
Mean is pulled by the outlier
Median is not
Mean is not a good center of these data

13. Mean vs. Median Skewness pulls the mean in the direction of the tail
Skewed to the right = mean > median
Skewed to the left = mean < median
Outliers pull the mean in their direction
Large outlier = mean > median
Small outlier = mean < median

14. Spread Range = maximum – minimum Thomas Sandberg
Min = 4, Max = 43, Range = = 39 HRs
Sandberg
Min = 0, Max = 40, Range = = 40 HRs

15. Spread Range is a very basic measure of spread
It is highly affected by outliers
Makes spread appear larger than reality
Ex. The annual numbers of deaths from tornadoes in the U.S. from 1990 to 2000:
Range with outlier: 130 – 25 = 105 tornadoes
Range without outlier: 94 – 25 = 69 tornadoes

16. Spread Interquartile Range (IQR) IQR = Q3 – Q1 First Quartile (Q1)
Larger than about 25% of the data
Third Quartile (Q3)
Larger than about 75% of the data
IQR = Q3 – Q1
Center (Middle) 50% of the values

17. Finding Quartiles Order the data Split into two halves at the median
When n is odd, include the median in both halves
When n is even, do not include the median in either half
Q1 = median of the lower half
Q3 = median of the upper half

18. Example – Frank Thomas Order the values (15 values)
Lower Half =
Q1 = Median of lower half = 21 HRs
Upper Half =
Q3 = Median of upper half = 40 HRs
IQR = 40 – 21 = 19 HRs

19. Example – Ryne Sandberg
Order the values (16 values)
Lower Half =
Q1 = Median of lower half = 8.5 HRs
Upper Half =
Q3 = Median of upper half = 26 HRs
IQR = Q3 – Q1 = 26 – 8.5 = 17.5 HRs

20. Five Number Summary
Minimum Q1
Median Q3
Maximum

21. Examples Thomas Sandberg Min = 4 HRs Q1 = 21 HRs Median = 32 HRs
Max = 43 HRs
Sandberg
Min = 0 HRs
Q1 = 8.5 HRs
Median = 17.5 HRs
Q3 = 26 HRs
Max = 40 HRs

22. Graph of Five Number Summary
Boxplot
Box between Q1 and Q3
Line in the box marks the median
Lines extend out to minimum and maximum
Best used for comparisons
Use this simpler method

23. Example – Thomas & Sandberg
Boxplot of Thomas Home Runs
Box from 21 to 40
Line in box 32
Lines extend out from box from 4 and 43
Boxplot of Sandberg Home Runs
Box from 8.5 to 26
Line in box at 17.5
Lines extend out from box to 0 and 40

24. Side by Side Boxplots of Thomas & Sandberg Home Runs

25. Spread Standard deviation “Average” spread from mean
Most common measure of spread
(Although it is influenced by skewness and outliers)
Denoted by letter s
Make a table when calculating by hand

26. Standard Deviation

27. Example – Deaths from Tornadoes53
=-3.27
10.69 39 =
298.25 33 =
541.49 69
= 12.73
162.05 30 =
690.11 25 =
977.81 67
= 10.73
115.13
130
= 73.73 94
= 37.73 40 =
264.71

28. Example – Frank Thomas
Find the standard deviation of the number of home runs given the following statistic:

29. Properties of s
s = 0 only when all observations are equal; otherwise, s > 0
s has the same units as the data
s is not resistant
Skewness and outliers affect s, just like mean
Tornado Example:
s with outlier: tornadoes
s without outlier: tornadoes

30. Which summaries should you use?
What numbers are affected by outliers?
Mean
Standard deviation
Range
What numbers are not affected by outliers?
Median
IQR

31. Which summaries should you use?
Five Number Summary
Skewed Data
Data with outliers
Mean and Standard Deviation
Symmetric Data
ALWAYS PLOT YOUR DATA!!