Statistics is a branch of Mathematics which deals with the collection, classification and representation of any kind of data to ease the analysis and understanding. Various forms of data representations in the statistics upto 10th level include bar graphs, pie charts, histograms and frequency polygons.Statistics upto 10th level also includes the computation of central tendencies of ungrouped random data. However, in statistics class 10 explanation, calculating central tendency of grouped data is demonstrated in detail.

Though statistics upto 10th level have a brief description of classifying the data into grouped frequency tables, the clear explanation of measuring its central tendency is stated in class 10 statistics notes. Central tendency is that value which represents the characteristics of the entire dataset considering each and every value in the set of data. The three measures of central tendency are Mean, Median and Mode. Class 10 statistics notes defines these measures of central tendency as follows:

Mean is the sum of all the observations in a data set divided by the total number of observations.

Mode of a grouped data is the observation with maximum frequency.

Median is that value which represents the middle-most observation in a data set.

Mean of a given data set can be found by three different methods namely:

### Direct Method

\[\overline x = \frac{{\sum {{f_i}} {x_i}}}{{\sum {{f_i}} }}\]

where is the mean of the grouped data

\[\underline {\sum {{f_i}{x_i}} } \]is the summation of the product of class mark and the corresponding frequency where ‘i’ may vary from 1 to n and ‘n’ is the total frequency

is the total frequency

### Assumed Mean Method:

This method is used when the class mark and frequency of grouped data is in several hundreds or considerably large.

\[\overline x = a + \frac{{\sum {{f_i}{d_i}} }}{{\sum {{f_i}} }}\]

where \[\overline x \] is the mean of the grouped data \[{x_i}\] ‘a’ is the assumed mean which is the value of \[{x_i}\] in the middle of the data.

\[\underline {\sum {{f_i}{d_i}} } \]is the summation of the product of class mark and the corresponding frequency deviation where ‘i’ may vary from 1 to n and ‘n’ is the total frequency. \[{d_i} = {\text{ }}{x_i} - a\]

\[\overline {\sum {{f_i}} } \] is the total frequency

### Step Deviation Method:

This method is used when the classmarks are the multiples of any common number.

\[\overline x = a + h\left( {\frac{{\sum {{f_i}{u_i}} }}{{\sum {{f_i}} }}} \right)\]

where \[\overline x \] is the mean of the grouped data

‘a’ is the assumed mean which is the value of \[{x_i}\] in the middle of the data.

\[{u_i} = \frac{{{x_i} - a}}{h}\]

where ‘i’ may vary from 1 to n and ‘n’ is the total frequency and ‘h’ is the class size

\[\overline {\sum {{f_i}} } \] is the total frequency

Median of a grouped set of data is calculated as:

\[Median = l + \left( {\frac{{\frac{n}{2} - cf}}{f}} \right) \times h\]

where ‘l’ is the lower limit of the median class

‘n’ is the number of observations

‘cf’ is the class preceding the median class

‘f’ is the frequency of median class

‘h’ is the class size

Median class is the class which has the cf value nearer to \[\frac{2}{n}\]

l = lower limit of median class, n = number of observations, cf = cumulative frequency of class preceding the median class, f = frequency of median class, h = class size (assuming class size to be equal).

Mode of a grouped frequency distribution is calculated using the formula

\[Mode = l\left( {\frac{{{f_i} - {f_o}}}{{2{f_i} - {f_o} - {f_2}}}} \right) \times h\]

where ‘l’ is the lower limit of the modal class

‘h’ is the class size which is assumed to be equal

‘f1’ is the frequency of the modal class

‘f0’ is the frequency of the class preceding the modal class

‘’f2’ is the frequency of the class succeeding the modal class

Modal class is the class with maximum frequency

Empirical formula statistics class 10 gives the relationship between the three measures of central tendency. The empirical formula is written as:

3 Median = Mode + 2 Mean

This formula can be used to measure the values of other central tendency two measures of central tendency are known.

Consider a grouped set of data which summarizes the scores of students of a class in Mathematics examination as shown in the table below.

Calculate the Following for the Data Described in the Above Table.

Mean by direct method

Mean by assumed mean method

Mean by step deviation method

Median

Mode

Verify the empirical formula

Solution:

Determination of Mean:

Each column is given a number for better understanding of the problem. Columns 1 and 2 are derived from the question.

Column 3 gives the class mark of each class interval which is calculated as

\[Class{\text{ }}Mark = \frac{{{\text{Upper class limit + Lower classs Limit}}}}{2}\]

Column 4 gives the product of xi and fi. This is used in calculating the direct mean.

\[Direct{\text{ }}mean = \frac{{\sum {fixi} }}{{\sum {fi} }} = \frac{{1860}}{{30}} = 62\]

meanColumn 5 is used to calculate the deviation of class mark from assumed mean. Here assumed mean a = 47.5. di is calculated as di= xi - a

Assumed Mean method is used to determine the mean as follows:

\[mean = a + \frac{{\sum {fidi} }}{{\sum {fi} }} = 47.5 + \frac{{435}}{{30}} = 47.5 + 14.5 = 62\]

Mean using step deviation method is calculated by determining \[{u_i}\] and computing \[{f_i}{u_i}\]

\[{u_i} = {\text{ }}\frac{{\left( {{x_i} - {\text{ }}a} \right)}}{h}{\text{ }} = {\text{ }}\frac{{{d_i}}}{h}\]It is shown in column 7 and 8. Class size in this example is

h = upper limit - lower limit = 15.

Mean by step deviation method is calculated as

\[mean = a + h\frac{{\sum {fiui} }}{{\sum {fi} }} = 47.5 + 15 \times \frac{{29}}{{30}} = 47.5 + 14.5 = 62\]

Determination of Median:

Calculating the cumulative frequency is done as follows

In the above table, cumulative frequency for more than type and less than type are found.

Now it is important to find the median class. Median class in the class with cf value near to \[\frac{n}{2}\]. Here \[\frac{n}{2} = \frac{{30}}{2} = 15\]. So the class with cf = 18 can be considered as median class.

So median class = 55 - 70. l = 55, f = 30, h = 15

Cf = 25 for more than type

\[Median = l + \left( {\frac{{\frac{n}{2} - cf}}{f}} \right) \times h\]

Median = 50 (More than type)

Determination of mode:

To calculate mode, modal class is to be identified. Modal class in the above example is 40 - 55.

So, l = 40, f1 = 7, f0 = 3, f2 = 6, h = 15

Substituting these values in the formula for mode of grouped data:

\[Mode = l\left( {\frac{{{f_i} - {f_o}}}{{2{f_i} - {f_o} - {f_2}}}} \right) \times h\]

Mode = 52

The values can also be verified using empirical formula statistics class 10 .

Central tendency measures help in simplifying data representations and comparing the data sets for better performances in statistics.

The median of a grouped data set can also be determined graphically by drawing more than type and less than type ogives. The point of intersection of the two ogive types gives the value of median on the ‘x’ axis.

FAQ (Frequently Asked Questions)

1. How can one analyze the choice of central tendency to be used according to statistics class 10 explanation?

As mean takes all the observations into consideration and its value lies between the extremes, this is the most frequently used measure of central tendency.

Mean also helps in comparison of two different sets of grouped data. However, the mean of a grouped data set varies with the changes in smallest and largest observations in the group.

In class 10 statistics explanation problems where individual values of each observation are not important, median can be used as a measure of central tendency.

Median can be used in applications such as computing productivity rate of workers and average wage in a company. The measure of median never gets affected by extreme values.

Mode can be used as a best measure of central tendency where the most frequent value or most popular brand is to be determined. In other words, mode gives the best choice among the group of observations. Example: Most frequently watched tv show, most likely color among girls etc.

2. Are the values of mean obtained by direct method, assumed mean method and step deviation method the same? Also compare the value with ungrouped data.

The mean obtained by an ungrouped set of data is the exact mean as it takes into account each and every value of the data.

However, in the method of determining the mean of the same data when it is grouped under class intervals, the mean obtained is an approximate value because it does not consider all the values of the data set. Instead, it considers the midpoint of a class interval.

The values of the mean of a grouped data set remains the same irrespective of the method used to determine the mean. The mean value remains the same for direct method, assumed mean method and step deviation method.