1. HS 67 - Intro Health Statistics Describing Distributions with Numbers
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Chapter 2
Describing Distributions with Numbers
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Chapter 2
BPS Chapter 2 1

2. Numerical Summaries of:
HS 67 - Intro Health Statistics
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Numerical Summaries of:
Central location
mean
median
Spread
Range
Quartiles
Standard Deviation / variance
Shape measures not covered
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3. HS 67 - Intro Health Statistics
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Arithmetic Mean
Most common measure of central location
Notation (“xbar”):
Where
n is the sample size
∑ is the summation symbol
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4. HS 67 - Intro Health Statistics
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Example: Sample Mean
Data: Metabolic rates, calories / day:
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5. HS 67 - Intro Health Statistics
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Median (M)
Half the values are less than the median, half are greater
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
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6. HS 67 - Intro Health Statistics
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Examples: Median
Example 1:
Median = 4
Example 2:
Median = 5 (average of 4 and 6)
Example 3:
Median  2
(Values must first be ordered first , Median = 4)
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7. HS 67 - Intro Health Statistics
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Example: Median
The location of the median in ordered array:
L(M) = (n + 1) / 2
Data = metabolic rates in slide 4 (n = 7)
Ordered array:
 median
Value of median = 1614
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8. The Median is robust to outliers
HS 67 - Intro Health Statistics
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The Median is robust to outliers
This data set:
has median 1614 and mean 1600
This similar data with high outlier:
still has median 1614 but now has mean
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9. The skew pulls the mean
The average salary at a high tech firm is $250K / year
The median salary is $60K
What does this tell you?
Answer: There are some very highly paid executives, but most of the workers make modest salaries, i.e., there is a positive skew to the distribution
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10. HS 67 - Intro Health Statistics
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Spread = Variability
Amount of spread around the center!
Statistical measures of spread
Range
Inter-Quartile Range
Standard deviation
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BPS Chapter 2 10

11. Range and IQR Range = maximum – minimum Easy, but NOT as good as the…
Quartiles & Inter-Quartile Range (IQR)
Quartile 1 (Q1) cuts off bottom 25% of data (“25th percentile”)
Quartile 2 (Q2) cuts off two-quarters of data
same as the Median!
Quartile 3 (Q3) cuts off three-quarters of the data (“75th percentile”)

12. HS 67 - Intro Health Statistics
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Obtaining Quartiles
Order data
Find the median
Look at the lower half of data set
Find “median” of this lower half
This is Q1
Look at the upper half of the data set.
Find “median” of this upper half
This is Q3
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13. Example: Quartiles Consider these 10 ages: 
median
The median of the bottom half (Q1) = 21 
The median of the top half (Q3) = 42 
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14. HS 67 - Intro Health Statistics
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Example 2: Quartiles, n = 53
Median = 165
L(M)=(53+1) / 2 = 27
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15. Bottom half has n* = 26  L(Q1)=(26 + 1) / 2= 13.5 from bottom
HS 67 - Intro Health Statistics
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Example 2: Quartiles, n = 53
Bottom half has n* = 26  L(Q1)=(26 + 1) / 2= 13.5 from bottom
Q1 = avg(127, 128) = 127.5
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16. Top half has n* = 26  L(Q3) = 13.5 from the top!
HS 67 - Intro Health Statistics
4/10/2017
Example 2: Quartiles, n = 53
Top half has n* = 26  L(Q3) = 13.5 from the top!
Q3 = avg(185, 185) = 185
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BPS Chapter 2 16

17. HS 67 - Intro Health Statistics
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11 009
14 08
16 555
19 245
20 3
21 025
22 0 23 24 25
26 0
Example 2 Quartiles
Q1 = 127.5
Q2 = 165
Q3 = 185
"5 point summary"
= {Min, Q1, Median, Q3, Max}
= {100, 127.5, 165, 185, 260}
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18. Inter-quartile Range (IQR)
HS 67 - Intro Health Statistics
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Inter-quartile Range (IQR)
Inter-Quartile
Range (IQR) = Q3  Q1
= 185 – = 57.5
Q1 = 127.5
Q3 = 185
“spread of middle 50%”
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19. Simple Box 5-point summary graphically
HS 67 - Intro Health Statistics
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Simple Box 5-point summary graphically Q1 M Q3
min
max
Weight
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20. Boxplots are useful for comparing groups
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21. Standard Deviation & Variance
HS 67 - Intro Health Statistics
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Standard Deviation & Variance
Most popular measures of spread
Each data value has a deviation, defined as:
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BPS Chapter 2 21

22. Example: Deviations Metabolic data (n = 7)
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23. HS 67 - Intro Health Statistics
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Variance
Find the mean
Find the deviation of each value
Square the deviations
Sum the squared deviations
Divide by (n − 1)
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24. HS 67 - Intro Health Statistics
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Data
Data: Metabolic rates, n = 7
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25. HS 67 - Intro Health Statistics
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“Sum of Squares”
Obs
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
11,200
214,870
SUMS
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26. HS 67 - Intro Health Statistics
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Variance
Sum of Squares
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27. HS 67 - Intro Health Statistics
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Standard Deviation
Square root of variance
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28. Standard Deviation Direct Formula
HS 67 - Intro Health Statistics
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Standard Deviation Direct Formula
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29. Use calculator to check work!
I’m supporting the TI-30XIIS only
TI-30XIIS sequence:
On > CLEAR > 2nd > STAT > Scroll > Clear Data > Enter
2nd > STAT > 1-VAR or 2-VAR
DATA > “enter data
STATVAR key

30. Choosing Summary Statistics
HS 67 - Intro Health Statistics
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Choosing Summary Statistics
Use the mean and standard deviation to describe symmetrical distributions & distributions free of outliers
Use the median and quartiles (IQR) to describe distributions that are skewed or have outliers
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31. Example: Number of Books Read
HS 67 - Intro Health Statistics
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Example: Number of Books Read
n = L(M)=(52+1)/2=26.5 M
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32. HS 67 - Intro Health Statistics
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Example: Books read, n = 52
5-point summary: 0, 1, 3, 5.5, 99 Highly asymmetric distribution
Number of books
The mean (“xbar” = 7.06) and standard deviation (s = 14.43) give false impressions of location and spread for this distribution and are considered inappropriate. Use the median and 5-point summary instead.
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