Numerical descriptors

BPS chapter 2

© 2006 W.H. Freeman and Company

Objectives (BPS chapter 2)

Describing distributions with numbers

Measure of center: mean and median

Measure of spread: quartiles and standard deviation

The five-number summary and boxplots

IQR and outliers

Choosing among summary statistics

Using technology

Organizing a statistical problem

The mean or arithmetic average

To calculate the average, or mean, add

all values, then divide by the number of

individuals. It is the “center of mass.”

Sum of heights is 1598.3

Divided by 25 women = 63.9 inches

58.2 64.059.5 64.560.7 64.160.9 64.861.9 65.261.9 65.762.2 66.262.2 66.762.4 67.162.9 67.863.9 68.963.1 69.663.9

Measure of center: the mean

n

nx....xxx

21

x 1598.3

2563.9

Mathematical notation:

x1

n ixi1

n

woman(i)

height(x)

woman(i)

height(x)

i = 1 x1= 58.2 i = 14 x14= 64.0

i = 2 x2= 59.5 i = 15 x15= 64.5

i = 3 x3= 60.7 i = 16 x16= 64.1

i = 4 x4= 60.9 i = 17 x17= 64.8

i = 5 x5= 61.9 i = 18 x18= 65.2

i = 6 x6= 61.9 i = 19 x19= 65.7

i = 7 x7= 62.2 i = 20 x20= 66.2

i = 8 x8= 62.2 i = 21 x21= 66.7

i = 9 x9= 62.4 i = 22 x22= 67.1

i = 10 x10= 62.9 i = 23 x23= 67.8

i = 11 x11= 63.9 i = 24 x24= 68.9

i = 12 x12= 63.1 i = 25 x25= 69.6

i = 13 x13= 63.9 n =25 =1598.3

Learn right away how to get the mean using your calculators.

Your numerical summary must be meaningful

Here the shape of the distribution is wildly irregular. Why?

Could we have more than one plant species or phenotype?

6.69x

The distribution of women’s height appears coherent and symmetrical. The mean is a good numerical summary.

3.69x

Height of 25 women in a class

Height of plants by color

0

1

2

3

4

5

Height in centimeters

Num

ber

of p

lants

red

pink

blue

58 60 62 64 66 68 70 72 74 76 78 80 82 84

A single numerical summary here would not make sense.

9.63x 5.70x 3.78x

Measure of center: the medianThe median is the midpoint of a distribution—the number such

that half of the observations are smaller and half are larger.

1. Sort observations from smallest to largest.n = number of observations

______________________________

1 1 0.62 2 1.23 3 1.64 4 1.95 5 1.56 6 2.17 7 2.38 8 2.39 9 2.510 10 2.811 11 2.912 3.313 3.414 1 3.615 2 3.716 3 3.817 4 3.918 5 4.119 6 4.220 7 4.521 8 4.722 9 4.923 10 5.324 11 5.6

n = 24 n/2 = 12

Median = (3.3+3.4) /2 = 3.35

3. If n is even, the median is the mean of the two center observations

1 1 0.62 2 1.23 3 1.64 4 1.95 5 1.56 6 2.17 7 2.38 8 2.39 9 2.510 10 2.811 11 2.912 12 3.313 3.414 1 3.615 2 3.716 3 3.817 4 3.918 5 4.119 6 4.220 7 4.521 8 4.722 9 4.923 10 5.324 11 5.625 12 6.1

n = 25 (n+1)/2 = 26/2 = 13 Median = 3.4

2. If n is odd, the median is observation (n+1)/2 down the list

Mean and median for skewed distributions

Mean and median for a symmetric distribution

Left skew Right skew

MeanMedian

Mean Median

MeanMedian

Comparing the mean and the median

The mean and the median are the same only if the distribution is

symmetrical. The median is a measure of center that is resistant to

skew and outliers. The mean is not.

The median, on the other hand,

is only slightly pulled to the right

by the outliers (from 3.4 to 3.6).

The mean is pulled to the

right a lot by the outliers

(from 3.4 to 4.2).

P

erc

en

t o

f p

eo

ple

dyi

ng

Mean and median of a distribution with outliers

4.3x

Without the outliers

2.4x

With the outliers

Disease X:

Mean and median are the same.

Mean and median of a symmetric distribution

4.3

4.3

M

x

Multiple myeloma:

5.2

4.3

M

x

and a right-skewed distribution

The mean is pulled toward the skew.

Impact of skewed data

M = median = 3.4

Q1= first quartile = 2.2

Q3= third quartile = 4.35

1 1 0.62 2 1.23 3 1.64 4 1.95 5 1.56 6 2.17 7 2.38 1 2.39 2 2.510 3 2.811 4 2.912 5 3.313 3.414 1 3.615 2 3.716 3 3.817 4 3.918 5 4.119 6 4.220 7 4.521 1 4.722 2 4.923 3 5.324 4 5.625 5 6.1

Measure of spread: quartiles

The first quartile, Q1, is the value in

the sample that has 25% of the data

at or below it.

The third quartile, Q3, is the value in

the sample that has 75% of the data

at or below it.

M = median = 3.4

Q3= third quartile = 4.35

Q1= first quartile = 2.2

25 6 6.124 5 5.623 4 5.322 3 4.921 2 4.720 1 4.519 6 4.218 5 4.117 4 3.916 3 3.815 2 3.714 1 3.613 3.412 6 3.311 5 2.910 4 2.89 3 2.58 2 2.37 1 2.36 6 2.15 5 1.54 4 1.93 3 1.62 2 1.21 1 0.6

Largest = max = 6.1

Smallest = min = 0.6

Disease X

0

1

2

3

4

5

6

7

Yea

rs u

nti

l dea

th

“Five-number summary”

Center and spread in boxplots

0123456789

101112131415

Disease X Multiple myeloma

Yea

rs u

ntil

deat

h

Comparing box plots for a normal and a right-skewed distribution

Boxplots for skewed data

Boxplots remain true

to the data and clearly

depict symmetry or

skewness.

IQR and outliers

The interquartile range (IQR) is the distance between the first and third quartiles (the length of the box in the boxplot)

IQR = Q3 - Q1

An outlier is an individual value that falls outside the overall pattern.

How far outside the overall pattern does a value have to fall to be considered an outlier?

Low outlier: any value < Q1 – 1.5 IQR

High outlier: any value > Q3 + 1.5 IQR

The standard deviation is used to describe the variation around the mean.

2

1

2 )(1

1xx

ns

n

i

1) First calculate the variance s2.

2

1

)(1

1xx

ns

n

i

2) Then take the square root to get

the standard deviation s.

Measure of spread: standard deviation

Mean± 1 s.d.

x

Calculations …

We’ll never calculate these by hand, so make sure you know how to get the standard deviation using your calculator.

2

1

1( )

1

n

is x xn

Mean = 63.4

Sum of squared deviations from mean = 85.2

Degrees freedom (df) = (n − 1) = 13

s2 = variance = 85.2/13 = 6.55 inches squared

s = standard deviation = √6.55 = 2.56 inches

Women’s height (inches)

Software output for summary statistics:

Excel—From Menu:

Tools/Data Analysis/

Descriptive Statistics

Give common

statistics of your

sample data.

Minitab

Choosing among summary statistics

Because the mean is not

resistant to outliers or skew, use

it to describe distributions that are

fairly symmetrical and don’t have

outliers.

Plot the mean and use the

standard deviation for error bars.

Otherwise, use the median in the

five-number summary, which can

be plotted as a boxplot.

Height of 30 women

58

59

60

61

62

63

64

65

66

67

68

69

Box plot Mean +/- sd

Hei

ght i

n in

ches

Box plot Mean ± s.d.

What should you use? When and why?

Arithmetic mean or median?

Middletown is considering imposing an income tax on citizens. City hall

wants a numerical summary of its citizens’ incomes to estimate the total tax

base.

In a study of standard of living of typical families in Middletown, a sociologist

makes a numerical summary of family income in that city.

Mean: Although income is likely to be right-skewed, the city government wants to know about the total tax base.

Median: The sociologist is interested in a “typical” family and wants to lessen the impact of extreme incomes.

Organizing a statistical problem

State: What is the practical question, in the context of a real-world setting?

Formulate: What specific statistical operations does this problem call for?

Solve: Make the graphs and carry out the calculations needed for this problem.

Conclude: Give your practical conclusion in the setting of the real-world setting.