1. Basic Practice of Statistics - 3rd Edition
Chapter 2
Describing Distributions with Numbers
BPS - 3rd Ed.
Chapter 2
Chapter 2

2. Basic Practice of Statistics - 3rd Edition
Numerical Summaries
Center of distribution
mean
median
Spread of distribution
five-point summary (& interquartile range)
standard deviation (& variance)
BPS - 3rd Ed.
Chapter 2
Chapter 2

3. Mean (Arithmetic Average)
Basic Practice of Statistics - 3rd Edition
Mean (Arithmetic Average)
Traditional measure of center
Notation (“xbar”):
Sum the values and divide by the sample size (n)
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4. Mean Illustrative Example: “Metabolic Rate”
Basic Practice of Statistics - 3rd Edition
Mean Illustrative Example: “Metabolic Rate”
Data: Metabolic rates, 7 men (cal/day) :
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Chapter 2

5. Basic Practice of Statistics - 3rd Edition
Median (M)
Half of the ordered values are less than or equal to the median value
Half of the ordered values are greater than or equal to the median value
If n is odd, the median is the middle ordered value
If n is even, the median is the average of the two middle ordered values
BPS - 3rd Ed.
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6. Basic Practice of Statistics - 3rd Edition
Median
Example 1 data:
Median (M) = 4
Example 2 data:
Median = 5 (average of 4 and 6)
Example 3 data:
Median  2
(order the values: , so Median = 4)
BPS - 3rd Ed.
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Chapter 2

7. Location of the Median L(M)
Basic Practice of Statistics - 3rd Edition
Location of the Median L(M)
Location of the median: L(M) = (n+1)÷2 , where n = sample size.
Example: If 25 data values are recorded, the Median is located at position (25+1)/2 = 13 in ordered array.
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Chapter 2

8. Median Illustrative Example
Basic Practice of Statistics - 3rd Edition
Median Illustrative Example
Data: Metabolic rates, n = 7:
L(M) = (7 + 1) / 2 = 4
Ordered array:
 median
Value of median = 1614
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Chapter 2
Chapter 2

9. Comparing the Mean & Median
Basic Practice of Statistics - 3rd Edition
Comparing the Mean & Median
Mean = median when data are symmetrical
Mean  median when data skewed or have outlier (mean ‘pulled’ toward tail) while the median is more resistant
If we switch this:
to this:
the median is still 1614 but the mean goes from 1600 to
BPS - 3rd Ed.
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Chapter 2

10. Question The average salary at a high tech company is $250K / year
The median salary is $60K.
How can this be?
Answer: There are some very highly paid executives, but most of the workers make modest salaries
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11. Basic Practice of Statistics - 3rd Edition
Spread = Variability
Variability  the amount values spread above and below the center
Can be measured in several ways:
range (rarely used)
5-point summary & inter-quartile range
variance and standard deviation
BPS - 3rd Ed.
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Chapter 2

12. Basic Practice of Statistics - 3rd Edition
Range
Based on smallest (minimum) and largest (maximum) values in the data set:
Range = max  min
The range is not a reliable measure of spread (affected by outliers, biased)
BPS - 3rd Ed.
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Chapter 2

13. Basic Practice of Statistics - 3rd Edition
Quartiles
Three numbers which divide the ordered data into four equal sized groups.
Q1 has 25% of the data below it.
Q2 has 50% of the data below it. (Median)
Q3 has 75% of the data below it.
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Chapter 2

14. Obtaining the Quartiles
Basic Practice of Statistics - 3rd Edition
Obtaining the Quartiles
Order the data.
Find the median
This is Q2
Look at the lower half of the data (those below the median)
The “median” of this lower half = Q1
Look at the upper half of the data
The “median” of this upper half = Q3
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Chapter 2

15. Illustrative example: 10 ages
AGE (years) values, ordered array (n = 10): |
   Q Q Q3
Q1 = 21
Q2 = average of 27 and 28 = 27.5
Q3 = 42
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16. Basic Practice of Statistics - 3rd Edition
Weight Data: Sorted n = 53  Median: L(M)=(53+1)/2=27  placing it at 165 L(Q1)=(26+1)/2=13.5  placing it between 127 and 128 (127.5) L(Q3) = 13.5 from the top  placing it between 185 and 185
Q1 = Q2 = 165 Q3 = 185
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Chapter 2

17. Weight Data: Quartiles
Basic Practice of Statistics - 3rd Edition
11 009
14 08
16 555
19 245
20 3
21 025
22 0 23 24 25
26 0
Weight Data: Quartiles
Q1 = 127.5
Q2 = 165
Q3 = 185
BPS - 3rd Ed.
Chapter 2
Chapter 2

18. Basic Practice of Statistics - 3rd Edition
Five-Number Summary
minimum = 100
Q1 = 127.5
M = 165
Q3 = 185
maximum = 260
Interquartile
Range (IQR) = Q3  Q1
= 57.5
IQR gives spread of middle 50% of the data
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Chapter 2

19. Basic Practice of Statistics - 3rd Edition
Boxplot
Central box spans Q1 and Q3.
A line in the box marks the median M.
Lines extend from the box out to the minimum and maximum.
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Chapter 2

20. Basic Practice of Statistics - 3rd Edition
Weight Data: Boxplot Q1 M Q3
min
max
Weight
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Chapter 2

21. Quartile extrapolation
Basic Practice of Statistics - 3rd Edition
Quartile extrapolation
Quartile divides data set into 4 segment: bottom, bottom middle, top middle, upper
With small data sets  extrapolate values
Illustrative data: 2, 4, 6, 8
2 | | | 8
Q Q Q3
Q1 = average of 2 and 4, which is 3
Q2 = average of 4 and 5, which is 5
Q3 = average of 6 and 8, which is 7
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Chapter 2

22. Boxplots  useful for comparing two groups (text p. 39)
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23. Variances & Standard Deviation
Basic Practice of Statistics - 3rd Edition
Variances & Standard Deviation
The most common measures of spread
Based on deviations around the mean
Each data value has a deviation, defined as
BPS - 3rd Ed.
Chapter 2
Chapter 2

24. Fig 2. 3: Metabolic Rate for 7 men, with their mean (
Fig 2.3: Metabolic Rate for 7 men, with their mean (*) and two deviations shown
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Chapter 2

25. Basic Practice of Statistics - 3rd Edition
Variance
Find the mean
Find the deviation of each value
Square the deviations
Sum the squared deviations: we call this the sum of squares, or SS
Divide the SS by n-1
(gives typical squared deviation from mean)
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Chapter 2

26. Basic Practice of Statistics - 3rd Edition
Variance Formula
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27. Standard Deviation Square root of the variance
Basic Practice of Statistics - 3rd Edition
Standard Deviation Square root of the variance
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28. Variance and Standard Deviation Illustrative Example
Basic Practice of Statistics - 3rd Edition
Variance and Standard Deviation Illustrative Example
Data: Metabolic rates, 7 men (cal/day) :
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Chapter 2
Chapter 2

29. Variance and Standard Deviation Illustrative Example (cont.)
Basic Practice of Statistics - 3rd Edition
Variance and Standard Deviation Illustrative Example (cont.)
Observations
Deviations
Squared deviations
1792
17921600 = 192
(192)2 = 36,864
1666
1666 1600 = 66
(66)2 = 4,356
1362
1362 1600 = -238
(-238)2 = 56,644
1614
1614 1600 = 14
(14)2 =
1460
1460 1600 = -140
(-140)2 = 19,600
1867
1867 1600 = 267
(267)2 = 71,289
1439
1439 1600 = -161
(-161)2 = 25,921
sum =
SS = 214,870
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30. Variance and Standard Deviation Illustrative Example (cont.)
Basic Practice of Statistics - 3rd Edition
Variance and Standard Deviation Illustrative Example (cont.)
Notes: (1) Use standard deviation s for descriptive purposes (2) Variance & standard deviation calculated by calculator or computer in practice
BPS - 3rd Ed.
Chapter 2
Chapter 2

31. Summary Statistics Two main measures of central location
Mean ( )
Median (M)
Two main measures of spread
Standard deviation (s)
5-point summary (interquartile range)
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Chapter 2

32. Choosing Summary Statistics
Basic Practice of Statistics - 3rd Edition
Choosing Summary Statistics
Use the mean and standard deviation for reasonably symmetric distributions that are free of outliers.
Use the median and IQR (or 5-point summary) when data are skewed or when outliers are present.
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Chapter 2

33. Example: Number of Books Read
Basic Practice of Statistics - 3rd Edition
Example: Number of Books Read
L(M)=(52+1)/2=26.5 M
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34. Illustrative example: “Books read”
Basic Practice of Statistics - 3rd Edition
Illustrative example: “Books read”
5-point summary: 0, 1, 3, 5.5, 99 Note highly asymmetric distribution
Number of books
“xbar” = s = The mean and standard deviation give false impression with asymmetric data
BPS - 3rd Ed.
Chapter 2
Chapter 2