deviations is computed. The plus and minus signs are
disregarded. The formula for computation of the
average deviation is as follows:
Average deviation =
Sd
n
where the Greek letter S (sigma) means the sum of d
(the deviations) and n is the number of items.
Standard Deviation
The standard deviation, like the average deviation,
is the measure of the scatter or spread of all values in a
series of observations. To obtain the standard deviation,
square each deviation from the arithmetic average of
the data. Then, determine the arithmetic average of the
squared deviations. Finally, derive the square root of
this average. This is also called the root-mean-square
deviation, since it is the square root of the mean of the
deviations squared.
The formula for computing standard deviation is
given as follows:
Standard deviation =
Sd
n
2
where d2 is the sum of the squared deviations from the
arithmetic average, and n is the number of items in the
group of data.
An example of the computations of average
deviation and standard deviation is given in table 6-1
and in the following paragraphs.
Table 6-1.—Computation of Average and Standard Deviation
January
year
Mean
temperature
Deviations
from mean
Deviations
squared
1978
47
– 4
16
1979
51
+ 0
0
1980
53
+ 2
4
1981
50
– 1
1
1982
49
– 2
4
1983
55
+ 4
16
1984
46
– 5
25
1985
52
+ 1
1
1986
57
+ 6
36
1987
50
– 1
1
Totals
Mean
510
51
26
2.6
104
3.2
Suppose, on the basis of 10 years of data
(1978-1987), you want to compute the average
deviation of mean temperature and the standard
deviation for the month of January. First, arrange the
data in tabular form (as in table 6-1). Given the year in
the first column, the mean monthly temperature in the
second column, the deviations from an arithmetic
average of the mean temperature in the third column,
and the deviations from the mean squared in the fourth
column.
To compute the average deviation:
1.
Add all the temperatures in column 2 and
divide by the number of years (10 in this case) to get the
arithmetic average of temperature.
2.
In column 3, compute the deviation from the
mean or average determined in step 1. (The mean
temperature for the 10-year period was 51°F.)
3.
Total column 3, disregarding the negative and
positive signs. (Total is 26.)
4.
Apply the formula for average deviation:
Sd
n
26
10
2.6 F
=
=
The average deviation of temperature during the
month of January for the period of record, 10 years, is
2.6°F.
To compute the standard deviation:
1.
Square the deviations from the mean (column
3).
2.
Total these squared deviations. In this case, the
total is 104.
3.
Apply the formula for standard deviation:
Standard deviation =
Sd
n
2
=
104
10
10
4
. = 3.225 or 3.2°F
The standard deviation of temperature for the
month and period in question is 3.2°F (rounded off to
the nearest one-tenth of a degree).
From the standard deviation just determined, it is
apparent that there is a small range of mean temperature
during January. If we had a frequency distribution of
temperature available for this station for each day of the
month, we could readily determine the percentage of
readings which would fall in the 6.4-degree spread (3.2
either side of the mean). From these data we could then
formulate a probability forecast or the number of days
6-5
