Lecture 4
Chapter 2. Numerical
descriptors
Objectives (PSLS Chapter 2)
Describing distributions with numbers

Measure of center: mean and median (Meas. Cent. Award)

Measure of spread: quartiles, standard deviation, IQR (Meas. Var. Award)

The five-number summary and boxplots (SUMS Award)

Dealing with outliers (outliers award)

Choosing among summary statistics (All Numeric Awards)

Organizing a statistical problem (Foundational)
Measure of center: the mean
The mean, or arithmetic average
To calculate the average (mean) of a data set, add all values, then
divide by the number of individuals. It is the “center of mass.”
x1  x 2  .... xn
x
n
n is the sample size
x is the variable
1 n
x   xi
n i 1
Measure of center: the median
The median is the midpoint of a distribution—the number such that
half of the observations are smaller, and half are larger.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
25
6.1
1) Sort observations from smallest to largest.
n = number of observations
2) The location of the median is (n + 1)/2 in the
sorted list
______________________________
If n is odd, the median
is the value of the
center observation
 n = 25
(n+1)/2 = 13
Median = 3.4
If n is even, the median
is the mean of the two
center observations
n = 24 
(n+1)/2 = 12.5
Median = (3.3+3.4)/2
= 3.35
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
Comparing the mean and the median
The median is a measure of center that is resistant to skew and
outliers. The mean is not.
Mean and median for a
symmetric distribution
Mean and median for
skewed distributions
Mean
Median
Left skew
Mean
Median
Mean
Median
Right skew
Measure of spread: quartiles
The first quartile, Q1, is the median
of the values below the median in the
sorted data set.
The third quartile, Q3, is the median
of the values above the median in the
sorted data set.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
0.6
1.2
1.6
1.9
1.5
2.1
2.3
2.3
2.5
2.8
2.9
3.3
3.4
3.6
3.7
3.8
3.9
4.1
4.2
4.5
4.7
4.9
5.3
5.6
6.1
Q1= first quartile = 2.2
M = median = 3.4
Q3= third quartile = 4.35
How fast do skin wounds heal?
Here are the skin healing rate data from 18 newts measured
in micrometers per hour:
28 12 23 14 40 18 22 33 26 27 29 11 35 30 34 22 23 35
Sorted data:
11 12 14 18 22 22 23 23 26 27 28 29 30 33 34 35 35 40
Median = ???
Quartiles = ???
Measure of spread: standard deviation
The standard deviation is used to describe the variation around the mean.
To get the standard deviation of a SAMPLE of data:
1) Calculate the variance s2
1 n
2
s 
(
x

x
)
 i
n 1 1
2
2) Take the square root to get the standard deviation s
1 n
2
s
(
x

x
)

i
n 1 1
Learn how to obtain the standard deviation of a sample using a spread sheet.
A person’s metabolic rate is the rate at which the body consumes energy.
Find the mean and standard deviation for the metabolic rates of a sample of 7 men
(in kilocalories, Cal, per 24 hours).
x   x1 / n  1600
2
(
x

x
)
 214,870
 i
df  n  1  6
s 2  (1 df ) ( xi  x ) 2
 214,870 6  35,811.7
s  35,811.7  189.2
*
Center and spread in boxplots
6.1
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
max = 6.1
Boxplot
7
Q3= 4.35
median = 3.4
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
5
4
3
2
1
Q1= 2.2
0
Disease X
min = 0.6
“Five-number summary”
Boxplots and skewed data
Years until death
Boxplots for a symmetric
and a right-skewed distribution
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Boxplots show
symmetry or
skew.
Disease X
Multiple Myeloma
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 a suspected outlier?

Suspected low outlier: any value < Q1 – 1.5 IQR

Suspected high outlier: any value > Q3 + 1.5 IQR
7.9
5.6
5.3
4.9
4.7
4.5
4.2
4.1
3.9
3.8
3.7
3.6
3.4
3.3
2.9
2.8
2.5
2.3
2.3
2.1
1.5
1.9
1.6
1.2
0.6
8
7
Q3 = 4.35
6
Years until death
25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
*
Distance to Q3
7.9-4.35 = 3.55
5
Interquartile range
Q3 – Q1
4.35-2.2 = 2.15
4
3
2
1
Q1 = 2.2
0
Disease X
Individual #25 has a survival of 7.9 years, which is 3.55 years
above the third quartile. This is more than 1.5  IQR = 3.225 years.
 Individual #25 is a suspected outlier.
Dealing with outliers: Baldi and Moore’s
Suggestions
What should you do if you find outliers in your data? It depends in part
on what kind of outliers they are:

Human error in recording information

Human error in experimentation or data collection

Unexplainable but apparently legitimate wild observations
 Are you interested in ALL individuals?
 Are you interested only in typical individuals?

Learn. Does the outlier tell you something interesting about
biology?
Don’t discard outliers just to make your data look better, and
don’t act as if they did not exist.
Choosing among summary statistics: B & M


Because the mean is not
resistant to outliers or skew, use
it is often used 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 and
the five-number summary, which
can be plotted as a boxplot.
Describe a distribution with its
S.U.M.S. (shape, unusual points,
middle, and spread).
Height of 30 women
69
68
67
Height in inches

66
65
64
63
62
61
60
59
58
Box plot
Boxplot
Mean +/Mean
± sd
s.d.
Deep-sea sediments.
Phytopigment concentrations in deep-sea sediments
collected worldwide show a very strong right-skew.

Which of these two values is the mean and which is the median?
0.015 and 0.009 grams per square meter of bottom surface

Which would be a better summary statistic for these data?