Download Describing the Spread: The Standard Deviation

Section 5.7
Describing the Spread:
The Standard Deviation
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
The most common numerical description of
a distribution is a combination of the mean
with the standard deviation.
The standard deviation and its close relative,
the variance, measure spread by looking at
how far the observations are from their
mean.
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
Standard Deviation “s” is a kind of “standard”
or average amount that the observed data
values deviate from the mean.
It is calculated by taking the square root of
the mean of the squared deviations, except
the mean is divided by (n – 1) instead of n.
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
The standard deviation of n observations
x1,x2,x3 , ,xn is
s
 x1  x 
2
  x2  x    x3  x  
2
2
n 1
  xn  x 
2
Example 1
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
Find the standard deviation of 7 purchase
prices :
3 4 5 7 10 12 15 (in dollars)
1. Find the mean.
3  4  5  7  10  12  15
x
 8 dollars
7
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
2.
Find the standard deviation.
3  8
s
2
  4  8    5  8    7  8   10  8   12  8   15  8 
2
2
2
2
2
7 1
 5    4    3    1   2   4    7 
2

2
2
2
6
25  16  9  1  4  16  49

6
120

 20  4.47 dollars
6
2
2
2
2
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
3.
Find the variance.
The variance is what is under the
square root of the standard deviation.
120 120
var iance 

 20
7 1
6
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
1.
2.
The variance is large if the observations
are widely spread around the mean.
The variance is small if the
observations are all close to the mean.
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
Properties of the standard deviation “s”:
1. “s” measures spread about the mean.
2. s = 0 only when there is no spread,
otherwise s > 0.
3. “s” has the same units of measurement
as the original observations
4. “s” is sensitive to extreme observations
or outliers
Example 2
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
Here are the systolic blood pressures of 10
randomly chosen adults:
147
141
120
124
127
132
98
112
120
128
Find the standard deviation of this data.
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
1.
Find the mean of the data.
147  141  120  124  127  132  98  112  120  128
x
10
1249
x
10
x  124.9
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
2.
Subtract the mean from each
observation.
147 – 124.9 = 22.1
132 – 124.9 = 7.1
141 – 124.9 = 16.1
98 – 124.9 = -26.9
120 – 124.9 = -4.9
112 – 124.9 = -12.9
124 – 124.9 = -0.9
120 – 124.9 = -4.9
127 – 124.9 = 2.1
128 – 124.9 = 3.1
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
3.
Square the number that you found in
#2.
(22.1)2 = 488.41
(7.1)2 = 50.41
(16.1)2 = 259.21
(-26.9)2 = 723.61
(-4.9)2 = 24.01
(-12.9)2 = 166.41
(-0.9)2 = 0.81
(-4.9)2 = 24.01
(2.1)2 = 4.41
(3.1)2 = 9.61
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
4.
Add the squared numbers and divide
by (10 – 1).
488.41  259.21  24.01  0.81  4.41  50.41  723.61  166.41  24.01  9.61
10  1
1750.9
variance 
9
variance  193.544
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
5.
Take the square root of the variance.
The answer will be the standard
deviation of the data.
s  194.544
s  13.948
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
We now have a choice between two
descriptions of the center and spread of a
distribution: the five-number summary or the
mean and standard deviation (s).
Here is how to choose the correct summary.
Chapter 5: Exploring Data: Distributions
Describing Spread: The Standard Deviation
Choosing a Summary
1. The five-number summary is usually
better than the mean and standard
deviation for describing a skewed
distribution or a distribution with
outliers.
2. Use the mean and standard deviation
only for reasonably symmetric
distributions with no outliers.