Measure of Central Tendency for Grouped Data – MEAN

Measure of central tendency for Grouped Data - Mean / Average

We will learn here to find one of the measures of Central tendency for Grouped Data called Mean or Average. Since, the data is grouped into classes it is not easy to locate Mean or Average accurately. But we can find approximate Mean by using Mean formulae.

Prerequisite / Revise this:

Mean Formula / Average Formula

Mean is defined as the ratio of sum of all observations to the number of observations.
Mean = \frac{Sum\,of\,observations}{Number\,of\,observations}

For an ungrouped frequency distribution, the Mean,
\bar{x}= \frac{\sum_{i=1}^n f_i x_i} {\sum_{i=1}^n f_i}

\sum_{i=1}^n f_i x_i = Sum of observations

\sum_{i=1}^n f_i = Total number of Observations

↪ In most of real life situations, data is usually so large that to make a meaningful study it needs to be condensed as grouped data.

↪ In Grouped frequency distribution, observations are classified into class intervals of same widths.

↪ By convention, the common observation belongs to the higher class, i.e., 10 belongs to the class interval 10-20 (and not to 0-10).

↪ The number of observations in each class is called Class frequency.

↪ It is assumed that the frequency of each class interval is centered around its mid-point. So the mid-point (or class mark) of each class can be chosen to represent the observations falling in the class.

Class-mark =\left[\frac {Upper\,limit + Lower\,limit}{2} \right]

Direct Method

The class marks serve as xi’s in this method. For the ith class interval, the frequency fi corresponds to the class mark xi.

Now, the mean can be computed using following mean formula

\bar{x}= \frac{\sum_{i=1}^n f_i x_i} {\sum_{i=1}^n f_i}

  • This method of finding the mean is known as the Direct Method.
  •  This method gives an approximate mean because of the mid-point assumption.

Remember: When this mean formula is used

(i) For Ungrouped frequency distribution,

xi = ith observation

fi = frequency of the ith observation.

(ii) For Grouped frequency distribution,

xi = class mark of the ith class interval

fi = frequency of the ith class interval.

Ex – Find the mean for given data

Measure of central tendency for Grouped Data - Mean / Average

Solution – We can write the given data in grouped frequency distribution table as following

Measure of central tendency for Grouped Data - Mean / Average

So, the mean x̄ of the given data is given by

\bar{x}=\frac{\sum_1{i=1}^n f_i x_i}{\sum_{i=1}^n f_i} \\ =\frac{1860}{30}=62

Assumed Mean Method

Sometimes when the numerical values of xi (class mark) and fi are large, finding the product of xi and fi becomes tedious and time consuming. We can’t change the fi’s, but we can change each xi to a smaller number, so that our calculations become easy. We can achieve this by subtracting a fixed number from each of these xi’s.

  • The first step is to choose one among the xi’s as the assumed mean, and denote it by ‘a’. We may take ‘a’ to be that xi which lies in the center of x_1, x_2, \cdots x_n.

So, in previous example, we can choose a = 47.5 or a = 62.5. Let us choose a = 47.5.

  • The next step is to find the difference between a and each of the xi’s, that is, the deviation (di) of ‘a’ from each of the xi’s i.e.,

d_i=x_i - a \\ =x_i - 47.5

  • The third step is to find the product of di with the corresponding fi, and take the sum of all the fidi’s (Σfidi).

Then the Mean of the deviations will be as:

\bar{d}= \frac{\sum_{i=1}^n f_i d_i} {\sum_{i=1}^n f_i}

  • Since in obtaining di, we subtracted ‘a’ from each xi, so, in order to get the mean ͞x , we need to add ‘a’ to d .

This can be explained mathematically as:

Mean of deviations,
\bar{a}= \frac{\sum_{i=1}^n f_i d_i} {\sum_{i=1}^n f_i}

= \frac{\sum_{i=1}^n f_i (x_i - a)} {\sum_{i=1}^n f_i}

= \frac{\sum_{i=1}^n f_i x_i} {\sum_{i=1}^n f_i} - \frac{\sum_{i=1}^n f_i a}{\sum_{i=1}^n f_i}

\bar{d}= \bar{x} - a\frac{\sum_{i=1}^n f_i }{\sum_{i=1}^n f_i}

\bar{x}= a + \bar{d}

\bar{x}= a + \frac{\sum_{i=1}^n f_i d_i} {\sum_{i=1}^n f_i}

Mean = Assumed Mean + Mean of deviations

Example: For previous example, we can write mean deviation table as following (a = 47.5)

Measure of Central Tendency for Grouped Data - Mean / Average

Substituting the values of a, Σfidi and Σfi from Table we get,

\bar{x} = 47.5+\frac{435}{30}

47.5 + 14.5 = 62

Therefore, the mean of the marks obtained by the students is 62.

Step-deviation method

↪ In previous example, if we find the mean by taking each of xi (i.e., 17.5, 32.5 and so on) as ‘a’, then the mean determined in each case will be the same, i.e., 62.

So, we can say that the the value of the mean obtained does not depend on the choice of ‘a’.

↪ We can also observe that deviations are common multiples of the class size i.e., the values in Column 4 are all multiples of 15. So, if we divide the values in the entire Column 4 by 15, we would get smaller numbers to multiply with fi. (Here, 15 is the class size of each class interval.)

↪ Let, u_i = \frac{x_i - a}{h}

where a is the assumed mean and h is the class size.

↪ Then, Mean of reduced deviations,

\bar{u} = \frac{\sum_{i=1}^n f_i u_i}{\sum_{i=1}^n f_i}

↪ Now, \bar {x} can be find as following

\bar{u} = \frac{\sum_{i=1}^n f_i u_i}{\sum_{i=1}^n f_i}

= \frac{\sum_{i=1}^n f_i (x_i - a)/h}{\sum_{i=1}^n f_i}

= \frac{1}{h} \left [\frac{\sum_{i=1}^n f_i x_i - a\sum_{i=1}^n f_i}{\sum_{i=1}^n f_i}\right]

= \frac{1}{h}\left[\frac{\sum_{i=1}^n f_i x_i}{\sum_{i=1}^n f_i}-a\frac{\sum_{i=1}^n f_i}{\sum_{i=1}^n f_i}\right]

= \frac{1}{h} \left[\bar{x} - a\right]

h\bar{u}=\left(\bar{x} - a\right)

\bar{x} = a + h\bar{u}

\bar{x} = a + h\frac{\sum_{i=1}^n f_i u_i}{\sum_{i=1}^n f_i}

Example: For the previous example, we can write the step deviation table as follow (a = 47.5)

Measure of Central Tendency for Grouped Data - Mean / Average

Now, substituting the values of a, h, Σfiui and Σfifrom the Table, we get
\bar{x}=47.5+15\times \frac{29}{30}

= 47.5 + 14.5 = 62

So, the mean marks obtained by a student is 62.

The method discussed above is called the Step-deviation method.

Note :

↪ the step-deviation method will be convenient to apply if all the di’s have a common factor (=h).

↪ The mean obtained by all the three methods is the same (an approximate mean).

↪ The assumed mean method and step-deviation method are just simplified forms of the direct method. Calculation is simplified by reducing xi.

↪ The choice of method to be used depends on the numerical values of xi and fi. If xi and fi are sufficiently small, then the direct method is an appropriate choice. If xi and fi are numerically large numbers, then we can go for the assumed mean method or step-deviation method. If the class sizes are unequal, and xi are large numerically, we can still apply the step-deviation method by taking h to be a suitable divisor of all the di’s.

↪ The formula x̄ = a + hū still holds if a and h are not as given above (i.e., a = xi & h = class size), but are any non-zero numbers such that ui = (xi − a)/h.

⏪ Measure of Central Tendency for Ungrouped Data Mode & Median for Grouped Data⏩

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