Introductory Statistics for
Laboratorians dealing with High
Throughput Data sets
Centers for Disease Control
Problem 7: Dispersion
• Prepare 2 line graphs, one for males and one
for females using the data presented below.
• Put both line graphs on the same axes.
Problem 7: Dispersion
Attitudes on Race Relations
Males
Females
X
f
X
f
9
1
9
1
8
1
8
1
7
3
7
0
6
3
6
3
5
4
5
10
4
3
4
2
3
2
3
2
2
2
2
0
1
1
1
1
Problem 7: Dispersion
12
10
8
Males
6
Females
4
2
0
0
1
2
3
4
5
6
7
8
9
Problem 7: Dispersion
12
10
8
Males
6
Females
4
2
0
1
2
3
4
5
6
7
8
9
Problem 7: Dispersion
• How can we quantify the difference between
the men and the women in this problem.
• Compute the mean (average) for the men.
• Compute the mean (average) for the women.
Problem 7: Dispersion
• What are the highest and lowest scores for the
men?
• What are the highest and lowest scores for the
women?
• Count the number of scores from lowest to
highest. This number is called the Range of the
scores.
• In this case the Range doesn’t help us describe
the difference between the males and the
females. We need better measures of dispersion.
Problem 8: Dispersion
• For the following data:
• What is the highest and lowest score?
• What is the Range? (count the number of scores
from the lowest to the highest.)
• What is the Mean (average)?
• How far is each person from the Mean? (Fill in
the column. Always subtract the mean from the
score. )
Problem 8: Dispersion
Data Table
N=
Subject
Score
X
Fred
0
George
1
Harry
2
Jerry
4
Larry
5
Jennifer
6
Jan
7
Joan
8
Jessica
8
Juana
9
Total =
Mean =
Distance from Mean
x = (Score – Mean)
Total deviation =
Squared Distance
from Mean
Sum Squares =
Problem 8: Dispersion
• Compute the “Sum of Squared Deviations
from the Mean” (SS) for this data set (or
sample or whatever you call it).
• Compute the variance of the sample.
• Compute the standard deviation of the
sample.
Dispersion Definitions
• The range is the number of scores from the
smallest to the largest.
• Deviation Score = Score – Mean
– Always subtract the mean from the score
– Always preserve the sign (positive or negative)
– The total of the deviation scores is always zero
• Sum Squares = Total of the squared deviation
scores. (SS)
• Variance = SS/N
• Standard Deviation = square root of variance
Standard Deviation
• Surely there is an easier way to measure
dispersion than using all this squaring and square
rooting.
• Turns out, the standard deviation is the exact
point on a normal curve where the second
derivative is zero.
• If you were skiing down the slope, it would get
steeper and steeper then it would start to flatten
out. That point is the standard deviation.
• That’s why it is the preferred measure of
dispersion.
Standard Deviation
Problem 9
• Given the following collection of scores: 2, 3,
5, 6, 6, 8
– Calculate the range of the scores
– Calculate the sum of squares
– Calculate the variance
– Calculate the standard deviation
Problem 9
Data Table
Subject
X
Fran
2
Frank
3
Frangelica
5
Fonz
6
Frieda
6
Fabiano
8
N=
Total =
Mean =
x2
Deviation score (x)
SS =
Normal distributions
Normal—or Gaussian—distributions are a family of
symmetrical, bell- shaped density curves defined by a mean m
(mu) and a standard deviation s (sigma): N (m, s).
1
f ( x) 
e
2 s
1  xm 
 

2 s 
2
x
e = 2.71828… The base of the natural logarithm
π = pi = 3.14159…
x
A family of density curves
Here the means are the same (m =
15) while the standard deviations
are different (s = 2, 4, and 6).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
Here the means are different
(m = 10, 15, and 20) while the
standard deviations are the same
(s = 3).
0
2
4
6
8
10
12
14
16
18
20
22
24
26
28
30
All Normal curves N (m, s) share the same properties

About 68% of all observations
Inflection point
are within 1 standard deviation
(s) of the mean (m).

About 95% of all observations
are within 2 s of the mean m.

Almost all (99.7%) observations
are within 3 s of the mean.
mean µ = 64.5
standard deviation s = 2.5
N(µ, s) = N(64.5, 2.5)
Reminder: µ (mu) is the mean of the idealized curve, while x is the mean of a sample.
σ (sigma) is the standard deviation of the idealized curve, while s is the s.d. of a sample.
Definitions: Statistical Symbols
• In an actual sample
– Scores are represented by
–
–
–
–
Mean = X
x
Deviation Score
Standard Deviation = s
Variance = s2
X
XX
• In a theoretical distribution (density curve)
– Mean = μ
– Standard Deviation = σ
– Variance = σ2