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Statistics For Class 10

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Last updated date: 22nd Mar 2024
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Statistics Class 10 Explanation

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 up to the 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. 


Class 10 Statistics Notes

Though statistics upto the 10th level has 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:

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

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

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


Determination of Mean, Median, and Mode of a Given Data Set

Mean:

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

  1. Direct Method

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


where \[ overline{x}\] is the mean of the grouped data,\[\sum f_{i}x_{i} \]  is the summation of the product of class marks and the corresponding frequency where ‘i’ may vary from 1 to n and ‘n’ is the total frequency.


  1. 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.


\[ {\sum {{f_i}{d_i}} } \]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. \[{d_i} = {\text{ }}{x_i} - a\]


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


  1. 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

\[ {\sum f_{i}}  \] is the total frequency


Median:

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 \[\frac{2}{n}\]


Mode:

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

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. 


Class 10 Statistics Notes: Example Problems

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.

  1. Mean by direct method

  2. Mean by assumed mean method

  3. Mean by step deviation method

  4. Median 

  5. Mode

  6. Verify the empirical formula 


Solution:


1

2

3

4

5

6

7

8

Class Interval (Score)

No of student 

(\[f\])

Class Mark 

\[\left( {{{\mathbf{x}}_{\mathbf{i}}}} \right)\]

\[\left( {{{\mathbf{x}}_{\mathbf{i}}}{{\mathbf{f}}_{\mathbf{i}}}} \right)\]

\[{d_i}\]

\[\left( {{{\mathbf{f}}_{\mathbf{i}}}{{\mathbf{d}}_{\mathbf{i}}}} \right)\]

\[\left( {{{\mathbf{u}}_{\mathbf{i}}}} \right)\]

\[\left( {{{\mathbf{f}}_{\mathbf{i}}}{{\mathbf{u}}_{\mathbf{i}}}} \right)\]

10 - 25

2

17.5

35.0

- 30

- 60

-2

-4

25 - 40

3

32.5

97.5

- 15 

-45

-1

-3

40 - 55

7

47.5

332.5

0

0

0

0

55 - 70 

6

62.5

375.0

15

90

1

6

70 - 85

6

77.5

465.0

30

180

2

12

85 - 100

6

92.5

555.0

45

270

3

18


\[\Sigma {f_i} = {\text{ }}30\]


\[\Sigma {f_i}{x_i} = 1860\]


\[\Sigma {f_i}{d_i} = {\text{ }}435\]


\[\Sigma {f_i}{d_i} = {\text{ }}29\]


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

Class Interval (Score)

No of student 

\[\left( {{{\mathbf{f}}_{\mathbf{i}}}} \right)\]

More than type

Cf 

Less than type

Cf

10 - 25

2

More than 10

30

Less than 25

2

25 - 40

3

More than 25

28

Less than 40

5

40 - 55

7

More than 40

25

Less than 55

12

55 - 70 

6

More than 55

18

Less than 70

18

70 - 85

6

More than 70

12

Less than 85

24

85 - 100

6

More than 85

6

Less than 100

30

 

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 .


Fun Facts

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.

FAQs on Statistics For Class 10

1. How can one analyse 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 the comparison of two different sets of grouped data. However, the mean of a grouped data set varies with the changes in the smallest and largest observations in the group. 


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


Median can be used in applications such as computing the productivity rate of workers and average wage in a company. The measure of the 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. For example, most frequently watched tv shows, most likely colour 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.