In Statistics, measurement of central tendency is initial attempt which provide the single value with assumption of calculated central value represents the entire data set. Central value helps to describe whole picture of obtain data. Central value is defining as a representative value which is obtain from given data set and supposed to represent the whole data set. There are different types of central value used in statistics which are explain below:

## Mean:

We generally understand mean as arithmetic mean which is obtain by summing all the values of data set and dividing the sum value by number of observation. Let the data are x_{1}, x_{2}, x_{3} ……., x_{n} then mean is calculate as

X¯= ( x_{1+}x_{2+}x_{3} …….+ x_{n} )/ n

Therefore, mean is calculating by using following formula:

X⎺= ∑x/n

It is the direct methods to calculate the mean where, X⎺ represent mean, ‘n’ represent number of observation, ‘X’ represent value of variable. The mean is also calculating by taking *assumed mean method* by using following formula:

X⎺= A + ∑(x+A)/n

Where, A= assumed mean

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*Consider the following data to calculate mean:*

*X: 2,4,6,8,10,12*

*Solution,*

*∑X= 2+4+6+8+10+12=42*

*n=6*

*Now,*

*Mean(X**⎺**)=∑x/n** = 42/6** =7*

*The mean is 7.*

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Consider the following data to calculate mean:

X | 2 | 4 | 6 | 8 | 10 | 12 |

f | 5 | 8 | 12 | 9 | 3 | 2 |

Solution,

X | f | fX |

2 | 5 | 10 |

4 | 8 | 32 |

6 | 12 | 72 |

8 | 9 | 72 |

10 | 3 | 30 |

12 | 2 | 24 |

N=39 | =240 |

Now,

Mean(X⎺) = ∑fX/N = 240/39 = 6.15

The mean is 6.15.

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Consider the following data to calculate mean:

X | 0-10 | 10-20 | 20-30 | 30-40 | 40-50 | 50-60 |

f | 11 | 15 | 23 | 19 | 12 | 7 |

Solution,

X | f | mid value (m) | fm |

0-10 | 11 | 5 | 55 |

10-20 | 15 | 15 | 225 |

20-30 | 23 | 25 | 575 |

30-40 | 19 | 35 | 665 |

40-50 | 12 | 45 | 540 |

50-60 | 7 | 55 | 385 |

N= 87 | ∑Fm= 2445 |

Now,

Mean(X⎺) = ∑Fm/N = 2445/87 = 28.1

The mean is 28.1

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## Median

It is another tool of measurement of central tendency which is use to calculate the middle value of the data. It divided the data into two equal parts. To calculated the median, at first data must be arrange in ascending order then, position is calculated. The median is obtaining or calculate. Let the data are x_{1}, x_{2}, x_{3} ……., x_{n }and x_{1} < x_{2} < x_{3} < ……. < x_{n} which ensure that the data are arrange in ascending order. Then, at first the position is located to find median. Following formula is used to calculate position of median:

Position of median= ((n+1)/2 )^{th} item for individual and discrete series

Position of median= (N/2 )^{th} item for continuous series

In case of continuous series, to obtain median another formula must be used which is shown below:

M_{d} =L + (N/2 – cf) i/f

Where, ‘L’ represent lower value of median class, ‘cf’ represent cumulative frequency less than n/2, ‘f’ represent frequency of median class & ‘i’ represent class interval

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*Consider the following data to calculate median:*

*X: 2 4 8 12 10 6*

*Solution,*

*Arranging the data in ascending order,*

*X: 2,4,6,8,10 &12*

*Now,*

*Position of median=(** (n+1)/2) ^{th} item*

* =((6+1)/2** ) ^{th} item = 3.5^{th} item*

*3.5 ^{th} item lies in 3^{rd} and 4^{th} item. So,*

*M _{d} =*

*(3rd item + 4th item)/2 =(6+8)/2*

*= 10*

*The median is 7.*

*————————————————————————————–*

*Consider the following data to calculate median:*

X |
2 |
4 |
6 |
8 |
10 |
12 |

f |
5 |
8 |
12 |
9 |
3 |
2 |

*Solution,*

X |
f |
CF |

2 |
5 |
5 |

4 |
8 |
13 |

6 |
12 |
25 |

8 |
9 |
34 |

10 |
3 |
37 |

12 |
2 |
39 |

N= 39 |

*Position of median=((N+1)/2** ) ^{th} item*

* =( (39+1)/2**) ^{th} item = 20^{th} item*

*20 ^{th} item lies in 25 in CF so, 6 is the Median of the series.*

*————————————————————————————–*

*Consider the following data to calculate median:*

X |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |

f |
11 |
15 |
23 |
19 |
12 |
7 |

* *

*Solution,*

X |
f |
CF |

0-10 |
11 |
11 |

20-Oct |
15 |
26 |

20-30 |
23 |
49 |

30-40 |
19 |
68 |

40-50 |
12 |
80 |

50-60 |
7 |
87 |

N=87 |

* *

*Position of median=(** N/2) ^{th} item*

* =( 87/2** ) ^{th} item = 43.5^{th} item*

*43.5 ^{th} item lies in 49 in CF so, 20-30 is the Median class in the series. So, *

*L= 20, F= 23, CF=26, i=10*

*M _{d} = L +(N/2 – cf) i/f *

* =20 +(43.5-26) x 10 /23*

* =27.61*

The median is 27.61

## Mode

Mode is another method that measure the central tendency of data and it is mostly used for qualitative variable in research but in statistics it is also useful for the quantitative variable. Mode is defining as maximum repetition number which is used as central value of data. There is no formula to calculate mode for individual and discrete series but for continuous series to calculate mode following formula is used after locating the model class:

M_{o} = L + ∆_{1 x i/(∆1+∆2) }

Where, L represent lower value of model class, ∆_{1} refer to difference from f_{1} to f_{0}, ∆_{2} represent difference from f_{1} to f_{2}, & i represent class interval.

————————————————————————————-

*Consider the following data to calculate mode:*

X |
2 |
4 |
6 |
8 |
10 |
12 |

f |
5 |
8 |
12 |
9 |
3 |
2 |

*Solution,*

X |
f |

2 |
5 |

4 |
8 |

6 |
12 |

8 |
9 |

10 |
3 |

12 |
2 |

* *

*From above table the value of x i.e. 6 has maximum repeated so, 6 is mode.*

*————————————————————————————–*

*Consider the following data to calculate mode:*

X |
0-10 |
10-20 |
20-30 |
30-40 |
40-50 |
50-60 |

f |
11 |
15 |
23 |
19 |
12 |
7 |

* *

*Solution,*

X |
f |

0-10 |
11 |

10-20 |
15 |

20-30 |
23 |

30-40 |
19 |

40-50 |
12 |

50-60 |
7 |

N=87 |

* *

*From above table the frequency of class 20-30 has highest frequency so, 20-30 is model class. *

*To calculate mode,*

*L=20, f _{0}=15, f_{1}=23, f_{2} =19 i= 10*

*M _{0} = L +*

*∆*

_{1 x i/(∆1+∆2)}* =20 +(23-15) x 10/(23-15+23-19)*

* =26.67*

The mode of given data is 26.67

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