1.2 Describing Distributions with Numbers

40-44 45-49 50-54 55-59 60-64 65-690

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4

6

8

10

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14

Ages of Presidents at Inauguration

Age at Inauguration

Num

ber

of P

resi

dent

s

Describe the Histogram in terms of center, shape, spread, and outliers???

Age of Presidents at Inauguration

President Age President Age President Age

Washington 57 Buchanan 65 Coolidge 51

J. Adams 61 Lincoln 52 Hoover 54

Jefferson 57 A. Johnson 56F. D,

Roosevelt 51Madison 57 Grant 46 Truman 60

Monroe 58 Hayes 54 Eisenhower 61J. Q. Adams 57 Garfield 49 Kennedy 43

Jackson 61 Arthur 51 L. Johnson 55Van Buren 54 Cleveland 47 Nixon 56

W. H. Harrison 68 B. Harrison 55 Ford 61Tyler 51 Cleveland 55 Carter 52Polk 49 McKinley 54 Reagan 69

Taylor 64 T. Roosevelt 42 Bush 64Fillmore 50 Taft 51 Clinton 46Pierce 48 Wilson 56 Bush 54

Harding 55 Obama 47

Mean:

The most common measure of center (A.K.A. average) Denoted by The Mean is considered Non-resistant because it is

sensitive to extreme values. May or may not be outliers. On Calculator use 1 Var Stat to get the mean.

Median:

The middle value of the set of data Denoted as M If the # of observations is odd, the median is the center

observation. If the # of observations is even then take the mean of the

two center observations. Median is resistant to extreme values On Calculator use 1 Var Stat to get the median.

Example 1: Find and M for the set of data

=41.3 M=34Example 2: Find and M for the set of data

=19.1 M=18.5Number of Hysterectomies performed by a female doctor in one year

5 7 10 14 18 19 25 29 31 33

Number of Hysterectomies performed by a male doctor in one year20 25 25 27 28 31 33 34 36 37 44 50 59 85 86

Comparison of and MIf……

◦ Symmetrical – then they are very similar (close in value)

◦ Skewed – Then is farther out in the tail than the median

◦ Exactly symmetrical – exactly the same

Measuring Spread: Range & the QuartilesRange = Largest Value – Smallest Value - Lower Quartile – median of the observations smaller than the median

- Median - Upper Quartile - median of the observations larger than the median

– Interquartile Range Outliers fall more than below or above

** 1 – Var stats on your Calculator gives them all to you.

5 – Number Summary

The 5# Summary consists of the smallest and largest observations from a set of data along with .

The 5# summary leads to a new graph called the box and whisker plot (boxplot).

Best used for comparing two sets of data

Example 3: Find any outliers for the set of data.

•

• Therefore, the observations 85 and 86 are both outliers for the set of data.

Number of Hysterectomies performed by a male doctor in one year20 25 25 27 28 31 33 34 36 37 44 50 59 85 86

Example 4: Create a boxplot for each set of data. What can you conclude?

Number of Hysterectomies performed by a female doctor in one year5 7 10 14 18 19 25 29 31 33

Min M Max

Number of Hysterectomies performed by a male doctor in one year20 25 25 27 28 31 33 34 36 37 44 50 59 85 86

Min M Max 18.5

Standard Deviation

Measures spread by looking at how far the observations are from the mean.

Denoted by s** 1 – Var stats / Sx

Properties of Standard Deviation

s measures spread about the mean and should be used only when the mean is used.

As s gets larger the observations are more spread out from the mean

s is highly influenced by outliers

Example 5: Find the standard deviation for the set of data

Number of Hysterectomies performed by a male doctor in one year20 25 25 27 28 31 33 34 36 37 44 50 59 85 86

𝑠=20.6

*** 5# Summary is usually better than the mean and standard deviation for describing a skewed distribution. Use the mean and standard deviation for data that is reasonably symmetric