Download Standard Deviation

Standard Deviation
This gives a better measure of spread than the range, which only looks at the biggest and smallest values.
The standard deviation is a measure of the average deviation of each data point from the mean value.
Standard deviation =
 (deviation from mean)2
Σf

=

Remember,
 ( x – mean )2
n
A better way to calculate this is to use:
Standard deviation =

 x2 - mean 2
n
If you have grouped data then
standard deviation =

 f x2 - mean 2
Σf
Note that you can calculate standard deviation using Excel
1
769857946
mean =  f x =  f x
n
Σf
Data set 1
Data set 2
Data set 3
Data set 4
Data set 5
Data set 6
6
1
1
4
4
4
6
6
1
5
6
4
6
6
2
6
6
4
7
7
11
7
7
9
7
7
12
8
7
9
7
12
12
9
9
9
Standard deviation =
square root of the variance
Variance =
Sum( x^2) /n - mean^2
x^2
Data set 1
Data set 2
Data set 3
Data set 4
Data set 5
Data set 6
36
1
1
16
16
16
36
36
1
25
36
16
36
36
4
36
36
16
49
49
121
49
49
81
Standard deviation =
Variance =
Data set 5
x - mean
deviation
squared
49
49
144
64
49
81
Mean
6.5
6.5
6.5
6.5
6.5
6.5
49
144
144
81
81
81
Sum of x^2
255
315
415
271
267
291
Median
6.5
6.5
6.5
6.5
6.5
6.5
variance
0.250
10.250
26.917
2.917
2.250
6.250
square root of the variance
sum of deviations squared / n
4
-2.5
6
-0.5
6
-0.5
7
0.5
7
0.5
9
2.5
6.25
0.25
0.25
0.25
0.25
6.25
2
mean =
Sum =
variance =
sd =
769857946
6.5
13.5
2.25
1.5
Mode
6
6
1
#N/A
6
4
Range
1
11
11
5
5
5
standard deviation
0.500
3.202
5.188
1.708
1.500
2.500
Sizes of holes bored in engine
castings
Sizes of holes bored in engine castings
Diameter
cm
cm
2.011
2.014
2.016
2.019
2.021
2.024
2.026
2.029
2.031
2.034
Frequency
7
16
23
9
5
60
Diameter
cm
cm
11
14
16
19
21
24
26
29
31
34
%
11.7
26.7
38.3
15
8.3
Frequency
7
16
23
9
5
60
Diameter
cm
2.011
2.016
cm
2.014
2.019
Frequency
7
16
2.021
2.026
2.031
2.024
2.029
2.034
23
9
5
60
thousandths cm
11
14
16
19
21
24
26
29
31
34
Frequency
7
16
23
9
5
60
midpoint
2.0125
2.0175
2.0225
2.0275
2.0325
fx
14.0875
32.28
f x^2
28.35109375
65.1249
46.5175
18.2475
10.1625
121.295
94.08164375
36.99680625
20.65528125
245.209725
median
=
LQ =
UQ =
IQ
Range =
mean =
sd =
2.0215833
0.0054384
mean =
21.583333
sd =
5.4384179
true mean
=
2.0215833
true sd =
0.0054384
Diameter
midpoint
12.5
17.5
22.5
27.5
32.5
fx
87.5
280
517.5
247.5
162.5
1295
f x^2
1093.75
4900
11643.75
6806.25
5281.25
29725
Data adjusted by subtracting 2, then multiplying by 1000
Obtain true mean by dividing by 1000, then adding 2
Obtain true sd by dividing by
1000
3
769857946
2.021978261
2.0175
2.023869565
0.006369565
Cum. Freq.
7
23
46
55
60
68% (approximately two thirds) of the data lies within 1 standard deviation of the mean;
approximately 95% of the distribution lies within 2 standard deviations of the mean;
approximately 99.5% of the distribution lies within 3 standard deviations of the mean.
The above statements apply to a Normal distribution but are approximately true for other data
which is symmetrical.
In our example above,
Mean = 2.0216 to 4 d.p.
s.d. = 0.0054 to 4 d.p.
Therefore approximately two thirds of the data lies in the range 2.0216 ± 0.0054
( i.e. between 2.0162 and 2.0270 cm)
Approximately 95% of the data lies in the range 2.0216 ± 0.0108
(i.e. between 2.0108 and 2.0324 cm).
Approximately 99.5% of the data lies in the range 2.0216 ± 0.0162
(i.e. between 2.0054 and 2.0378 cm).
From the original table of data, 80% of the data lies between 2.015 and 2.030 cm.
Alternative measures of spread: Centiles and the interquartile range
Cumulative frequency
70
60
50
40
30
20
10
0
2.005
2.01
2.015
2.02
2.025
2.03
2.035
2.04
Diameter (cm)
Use the cumulative frequency curve to estimate the lower quartile, upper quartile, 5 th
centile and 95th centile.
4
769857946