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Find the mean of the following data by using step-deviation method

Class10-1415-1920-2425-2930-34
Frequency1511013511525


Answer
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Hint: First of all, the mean of any data using standard deviation method can be given by the formula, Mean,X=l+i=1nfiuii=1nfi×h. Now, we will find the values of summation of frequencies, summation of product of fi and ui by making a table of distribution and then based on the values of that table and summation we will find our answer.

Complete step-by-step answer:
First of all, we will convert this table into vertical and based on the formula of mean i.e. Mean,X=l+i=1nfiuii=1nfi×h, where, l is middle value of the given data that is assumed as mean for calculation, h is class size, so the table can be given as,

ClassFrequency (fi)
10-1415
15-19110
20-24135
25-29115
30-3425


Now, xi is the middle term in each class such as, to find the value of xi let’s consider the class 10-14, so the terms in 10-14 are 10, 11, 12, 13 and 14. So, xi will be 12. In the same way for class 15-19, xi will be 17, for 20-24 it will be 22, for 25-29 it will be 27 and for 30-34 it will be 32. So, we will substitute the values in the table.

ClassFrequency (fi)xi
10-141512
15-1911017
20-2413522
25-2911527
30-342532


Now, di is the difference between xi and the middle term of the terms of xi i.e. 22, which can be seen mathematically as,
di=xi22
Now, substituting the values of xi related to each class we will find the value of di, so, for class 10-14, value of di will be,
di=1222=10
Same way, for class 15-19, value of di will be, 1722=5, for class 20-24 value of di will be, 2222=0, for class 25-29, value of di will be, 2722=5 and for class 30-34 value of di will be, 3222=10. So, now, we will substitute these values in table so the table will be as,

ClassFrequency (fi)xidi
10-141512-10
15-1911017-5
20-24135220
25-29115275
30-34253210


Now, value of ui can be given by the formula, ui=di5, now, on substituting the values of di we will get values of ui as,
For, di=10, ui=105=2
For, di=5, ui=55=1
For, di=0, ui=05=0
For, di=5, ui=55=1
For, di=10, ui=105=2
On substituting these values, we will get table as,

ClassFrequency (fi)xidiui=di5
10-141512-10-2
15-1911017-5-1
20-241352200
25-291152751
30-342532102


Now, summation of frequencies can be given as,
fi=15+210+135+115+25=400
Now, product of fi and ui can be given as,

ClassFrequency (fi)xidiui=di5fiui
10-141512-10-2-30
15-1911017-5-1-110
20-2413522000
25-291152751115
30-34253210250


Now, summation of product of fi and ui can be given as
fiui=30110+0+115+50=140+165
fiui=25
Now, equation of mean can be given as,
Mean,X=l+i=1nfiuii=1nfi×h
Where, h=5 as the difference between the size of class for all the classes is 5, which can be calculated by reducing 0.5 from the value of class then taking their subtraction which can be given as 13.59.5=5, and l=22. On substituting all the values in equation, we will get,
X=22+25400×5
X=22+25×5400=22+0.3125
X=22+0.3125=22.3125
Hence, the mean of the given data using step deviation method is 22.3125.

Note: Here, in question it was mentioned to find the mean by step deviation method so we solved the problem using the formula, Mean,X=l+i=1nfiuii=1nfi×h. Now, students might use the formula, Mean,X=i=1nfixii=1nfi, where, product of fi and xi for each class will be,

ClassFrequency (fi)xifixi
10-141512225
15-19110171870
20-24135222970
25-29115273105
30-342532800


So, the summation of product of fi and xi will be, fiui=225+1870+2970+2530+675=, on substituting the values we will get, X=8225400=22.425 . As we can see that the answer remains the same with minute variation in it, so, we can consider this method as an alternative method to the problem.

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