Mean

Mean or average or arithmetic mean is one of the representative values of data. We can find the mean of observations by dividing the sum of all the observations by the total number of observations.

Mean of raw data:

If x1, x2, x3, ……. xn are n observations, then

Arithmetic Mean = (x1, x2, x3, ……. xn)/n

= (∑xi)/n

∑ (Sigma) is a Greek letter showing summation


1. Weights of 6 boys in a group are 63, 57, 39, 41, 45, 45. Find the mean weight.

Solution:

Number of observations = 6

Sum of all the observations = 63 + 57 + 39 + 41 + 45 + 45 = 290

Therefore, arithmetic mean = 290/6 = 48.3


Mean of tabulated data:

If x1, x2, x3, x4, ……. xn are n observations, and f1, f2, f3, f4, ……. fn represent frequency of n observations.

Then mean of the tabulated data is given by

= (f1 x1 + f2 x2 + f3 x3 + ……. fn xn)/(f1 + f2 + f3 + …… fn) = ∑(fixi)/∑fi


2. A die is thrown 20 times and the following scores were recorded 6, 3, 2, 4, 5, 5, 6, 1, 3, 3, 5, 6, 6, 1, 3, 3, 5, 6, 6, 2.

Prepare the frequency table of scores on the upper face of the die and find the mean score.

Solution:

Number on the upper
face of die
Number of times it occurs
(frequency)
fixi
1 2 1 × 2 = 2
2 2 2 × 2 = 4
3 5 3 × 5 = 15
4 1 4 × 1 = 4
5 4 5 × 4 = 20
6 6 6 × 6 = 36


Therefore, mean of the data   = ∑(fixi)/∑fi

                                        = (2 + 4 + 15 + 4 + 20 + 36)/20

                                        = 81/20

                                        = 4.05



3. If the mean of the following distribution is 9, find the value of p.

X 4 6 p + 7 10 15
f 5 10 10 7 8


Solution:

Calculation of mean

xi fi xifi
4 5 20
6 10 60
p + 7 10 10(p + 7)
10 7 70
15 8 120


∑fi = 5 + 10 + 10 + 7 + 8 = 40

∑ fixi = 270 + 10(p + 7)

Mean = ∑(fixi)/∑fi

9 = {270 + 10(p + 7)}/40

⇒ 270 + 10p + 70 = 9 × 40

⇒ 340 +10p = 360

⇒ 10p = 360 - 340

⇒ 10p = 20

⇒ p = 20/10

⇒ p = 2



Mean of grouped data:

While calculating the mean of the grouped data, the values x1, x2, x3, ……. xn are taken as the mid-values or the class marks of various class intervals. If the frequency distribution is inclusive, then it should be first converted to exclusive distribution.


4. The following table shows the number of plants in 20 houses in a group

Number of Plants 0 - 2 2 - 4 4 - 6 6 - 8 8 - 10 10 - 12 12 - 14
Number of Houses 1 2 2 4 6 2 3

Find the mean number of plans per house

Solution:

We have
   
Number of Plant Number of Houses
(fi)
Class Mark
(xi)
fixi
0 - 2 1 1 1 × 1 = 1
2 - 4 2 3 2 × 3 = 6
4 - 6 2 5 2 × 5 = 10
6 - 8 4 7 4 × 7 = 28
8 - 10 6 9 6 × 9 = 54
10 - 12 2 11 2 × 11 = 22
12 -14 3 13 3 × 13 = 39


∑fi = 1 + 2 + 2 + 4 + 6 + 2 + 3 = 20

∑fi xi =1 + 6 + 10 + 28 + 54 + 22 + 39 = 160

Therefore, mean = ∑(fixi)/ ∑fi = 160/20 = 8 plants

Statistics













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