MEAN
If any data set consisting of the values a1, a2, a3, …., an then the arithmetic
mean is defined as
Mean = Sum of all observation
If these n observations have corresponding frequencies, the arithmetic mean is
computed using the formula
a1+ a2 + a3+ ….+ an
Find the mean of the following data:
10, 15, 12, 16, 15, 10, 14, 15, 12, 10.
Solution:
Here data are repeated in the collection, so we take note of their frequencies.
Therefore, mean A =
= 10*3 + 12*2 +14*1 +15*3 +16*1
=
= 12.9
Formula of Arithmetic mean for grouped data
Calculate the mean of the following distribution:
Class Interval | ||||||
Frequency | 8 | |
class | Mid value xi) | ||
0-10 | 5 | 10 | 50 |
10-20 | 15 | 20 | 300 |
20-30 | 25 | 12 | 300 |
30-40 | 35 | 35 | 1225 |
40-50 | 45 | 24 | 1080 |
50-60 | 55 | 18 | 990 |
Total | 100 | 3945 |
And substituting this in direct method we will get the formula of assumed mean method. Assumed mean method is given by
Mean = a +
where , a = assumed number
d = x - a
Examples:
Find the mean of the following data, by using the assumed mean method.
Class Interval | |||||
Frequency |
Solution:
Class interval Frequency ()
x f
5 -20 -140 15 - -80 0 0 35 10 130 20 200
Mean = a +
(iii) Step deviation method: This method is mainly used when the deviation of the class from the assumed
mean are large and they all have a common factor. In this method, we use class width as ' h ' also . Using the same formula in assumed mean d = x-a here we use class width u =x-a h Step deviation formula is given as: Mean = a + h ( where , h = class width a= assumed number Example:
Class interval | Frequency () | ||
x | f | ||
5 | -20 | -140 | |
15 | - | -80 | |
0 | 0 | ||
35 | 10 | 130 | |
20 | 200 | ||
Find the mean marks of students from the following cumulative frequency distribution:
|
Marks | Number of students |
0-10 | 3 |
10-20 | 5 |
20-30 | 7 |
30-40 | 10 |
40-50 | 12 |
50-60 | 15 |
60-70 | 12 |
70-80 | 6 |
80-90 | 2 |
90-100 | 8 |
Now, we compute mean by the method of step deviation
Marks | Mid-value | Frequency | ||
0-10 | 5 | 3 | -5 | -15 |
10-20 | 15 | 5 | -4 | -20 |
20-30 | 25 | 7 | -3 | -21 |
30-40 | 35 | 10 | -2 | -20 |
40-50 | 45 | 12 | -1 | -20 |
50-60 | 55 | 15 | 0 | 0 |
60-70 | 65 | 12 | 1 | 12 |
70-80 | 75 | 6 | 2 | 12 |
80-90 | 85 | 2 | 3 | 6 |
90-100 | 95 | 8 | 4 | 32 |
Total | | = -26 |
=
,
a= 55 and h= 10
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