1. Describing Distributions Numerically
Chapter 3
Describing Distributions Numerically

2. Describing the Distribution
Center
Median
Mean
Spread
Range
Interquartile Range
Standard Deviation

3. Median Literally = middle number (data value)
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 – August Temps 13 observations
High Temperatures for Des Moines, Iowa taken from the first 13 days of August 2005.
Remember to order the values, if they aren’t already in order!
13 observations
(13+1)/2 = 7th observation from the bottom
Median = 90

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 – Yankees 10 observations
(10 + 1)/2 = 5.5, average of 5th and 6th observations from bottom
Median = 5
Scores of last 10 games
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
Formula
n observations
y1, y2, y3, …, yn are the values

8. Mean ( )

9. Example – Vikings (as of 1/9)
Find the mean of the (17 values)

10. Example – Colts as of (1/9)
Find the mean of the scores (17 values)

11. 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

12. 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

13. 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

14. 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

15. Weighted Mean Used when values are not equally represented.

16. Example (weighted mean)
A recent survey of new diet cola reported the following percentages of people who liked the taste. Find the weighted mean of the percentages.
Area
% Favored
Number surveyed 1 40
1000 2 30
3000 3 50
800

17. Example (cont.)
x1 = .40 x2 = .30 x3 = .50 w1 = 1000 w2 = 3000 w3 = 800
Use formula: {.40(1000) + .30(3000) + .50(800)} / { } = 1700/4800 = = 35.4%

18. Spread Range is a very basic measure of spread (Max – Min).
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
Range without outlier: 94 – 25 = 69

19. Spread Interquartile Range (IQR) IQR = Q3 – Q1 First Quartile (Q1)
25th Percentile
Third Quartile (Q3)
75th Percentile
IQR = Q3 – Q1
Center (Middle) 50% of the values

20. 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

21. Top 15 Populations US Cities 2004
New York, N.Y.
810
Los Angeles, Calif.
385
Chicago, Ill.
286
Houston, Tex.
201
Philadelphia, Pa.
147
Phoenix, Ariz.
142
San Diego, Calif.
126
San Antonio, Tex.
124
Dallas, Tex.
121
San Jose, Calif. 90
Detroit, Mich.
Indianapolis, Ind. 78
Jacksonville, Fla.
San Francisco, Calif. 74
* Populations were all divided by 10,000.

22. Example – Top City Populations
Order the values (14 values)
Lower Half = Q1 = Median of lower half = 90
Upper Half =
Q3 = Median of upper half = 201
IQR = Q3 – Q1 = = 111

23. August High Temps (8/1–8/13)
Order the values (13 values)
Lower Half =
Q1 = Median of lower half = 81
Upper Half =
Q3 = Median of upper half = 93
IQR = Q3 – Q1 = = 12

24. August High Temps (8/14–8/25)
Order the values (12 values)
Lower Half =
Q1 = Median of lower half = 78
Upper Half =
Q3 = Median of upper half = 87
IQR = Q3 – Q1 = = 9

25. Five Number Summary
Minimum Q1
Median Q3
Maximum

26. Examples Vikings (as of 1/9) Colts (as of 1/9) Min = 13 Q1 = 20
Median = 27
Q3 = 31
Max = 38
Colts (as of 1/9)
Min = 14
Q1 = 24
Median = 34
Q3 = 41
Max = 51

27. 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

28. Example – Vikings & Colts
Boxplot of Vikings scores
Box from 20 to 31
Line in box 27
Lines extend out from box from 14 and 38
Boxplot of Colts scores
Box from 24 to 41
Line in box at 34
Lines extend out from box to 14 and 51

29. Side by Side Boxplots of Vikings Scores and Colts Scores

30. Spread Standard deviation “Average” spread from mean
Most common measure of spread
Denoted by letter s
Make a table when calculating by hand

31. Standard Deviation

32. 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

33. Example - Vikings
Find the standard deviation of the scores of Vikings games given the following statistic:

34. 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: 31.97
s without outlier:

35. Which summaries should you use with different distributions?
The appropriate measures of center and spread when your distribution is symmetric are:
Mean
Standard deviation
The appropriate measures of center and spread when your distribution is skewed are:
Median
IQR

36. Comparing Variance
When comparing the variance for two sets of numbers find the coefficient of variation:
Formula = Cvar = =
Then compare the percentages.

37. Standardizing (first look)
I got a 85 on my English test and you got a 36 on your Spanish test. Who did better?
How can we compare things that come from different scales?
Standardizing
Use z formula (called z-score)

38. Standardizing Z=standardized score X = raw score
X-bar = mean of raw scores
S = sample standard deviation
So what does this mean for our test scores?

39. Standardizing
I got a 85 on my English test and you got a 35 on your Spanish test. Who did better?
Now I need to give you more information.
The English class’s tests had a mean of 83 and a standard deviation of 3.
The Spanish tests had a mean of 30 and a standard deviation of 2.

40. Standardizing

41. Comparing Standardized Scores
I scored .667 standard deviations above the mean on my English test where you scored 2.5 standard deviations above the mean on your Spanish test.
Comparatively you scored better on your exam.