MEASURES OF CENTRAL TENDENCY



MEASURES OF CENTRAL TENDENCY

Measures of central tendency refer to all those methods of statistical analysis by which averages of the statistical series are worked out.

The word average is very commonly used in day-to-day conversation. However, in statistics the term average has a different meaning. It is defined as that value of a distribution, which is considered as the most representative value of the series. Since an average represents the entire data, its value lies between the largest and smallest item. For this reason, an average is frequently referred to as a measure of central tendency.

Functions of averages

Gives one single value that describes the characteristics of the entire data.

Facilitates comparison

Requisites of a good average

It should be easy to understand

It should be simple to compute

It should be based on all observation

It should be capable of further algebraic treatment

Types of averages

Arithmetic mean (simple and weighted)

Median

Mode

Arithmetic mean (simple)

The most popular and widely used measure for representing the entire data by one value is what most laymen call as average and what statisticians call as arithmetic mean

Calculation of arithmetic mean in individual series (ungrouped data)

Direct method

Steps

• Find the sum of all the variables( ∑X)

• Divide the sum by the no. of items (N)

• Apply the formula: AM = ∑X ∕ N

Short cut method

• Take one of the given variables as the assumed mean (preferably the one in the middle) and denote it as ‘a’

• Find the deviation of the given variables from the assumed mean, and denote it as d (ie; d = x-a )

• Find the sum of deviations

• Apply the formula A.M = A + ∑(X-A) ∕ N

Calculation of arithmetic mean in discrete series (grouped data)

Direct method

Steps

• Find the product of the variable and the corresponding frequency (fx)

• Find the sum of fx ie; (∑fx)

• Find the sum of frequencies ie;( ∑f )

• Apply the formula: AM = ∑(fx) ∕ ∑f

Short cut method

• Take one of the given variables as the assumed mean (preferably the one in the middle) and denote it as ‘A’

• Find the deviation of the given variables from the assumed mean, X-A.

• Multiply the deviations with their corresponding frequencies f(X-A)

• Find the sum of f(X-A)

• Apply the formula A.M = A+ ∑(f(X-A) ∕ ∑f

Step deviation method

• Take one of the given variables as the assumed mean (preferably the one in the middle) and denote it as ‘A’

• Find the deviation of the given variables from the assumed mean, and denote it as (X-A)

• Divide the deviations by a common factor ( c ) and denote it as (X-A)’

• Multiply the deviations with corresponding frequencies and find the sum

i.e. ∑(f(X-A)’

• Apply the formula. A.M = A+ ∑(f(X-A)’ ∕ ∑f x C

Numerical questions

1. Find the value of arithmetic mean from the following using direct, short cut and step deviation method ans; 34

|Serial No. of students|1 |2 |3 |4 |5 |6 |

|No.ofstudents |13 |8 |15 |12 |10 |2 |

4. Calculate A.M from the following using all the three methods ans; 34.25

|income |20 |25 |30 |35 |40 |45 |50 |

|No. of persons |5 |8 |12 |15 |10 |6 |4 |

5. Find the value of mean from the following distribution ans; 33.6

|class interval 0-10 10-20 20-30 30- 40 40- 50 50-60 60-70 |

|frequencies 8 12 14 16 15 9 6 |

6. Calculate mean from the following ans;23.8

|Mid value 5 10 15 20 25 30 35 40 45 |

|Frequency 7 11 18 29 32 27 16 9 1 |

7. From the following frequency distribution find the value of missing frequency, if the value of mean is 12 ans 24

|class interval 0-4 4-8 8- 12 12- 16 16- 20 20- 24 24-28 |

|Frequency 4 4 ? 10 4 4 2 |

Weighted arithmetic mean

In simple arithmetic mean, all items are given equal importance. In weighted arithmetic mean, items are accorded different weights according to their relative importance.

Calculation of weighted arithmetic mean

• Items are multiplied by their corresponding weights (w) and summed up to get ∑XW

• ∑XW is divided by the sum total of weights

• Weighted arithmetic mean = ∑XW / ∑W

Properties of arithmetic mean

The sum of the deviations taken from arithmetic mean is always zero

The sum of the squares of deviations derived from arithmetic mean will be lesser than the sum of deviations derived from any other value

Partition values

Median

Median is the middle value of series when arranged in ascending or descending order. In the words of connor “ median is the value of the variable which divides the group into two equal parts, one part comprising of all values greater than and the other values less than median”

Calculation of median

1. Individual series

• Arrange the data in ascending or descending order

• Apply the formula median (M) = size of (N+1) / 2 th item, (N is the number of items in the series)

2. Discrete series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find (N+1) /2, (N is the total frequency)

• Locate the total which is equal to or greater than (N+1) /2 in the cumulative frequency column and determine the value of the variable corresponding to this to get the value of median.

3. Continuous series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find (N) /2, (N is the total frequency) in the cumulative frequency column and determine the median class

• Apply the following formula to find the value of median

Median = L +( N/2 – C.f) / f x i

In the above formula L = lower limit of the median class

N = total frequency

Cf = cumulative frequency of the class above the median class

I = class width of the median class

F = is the frequency of the median class

LOCATION OF MEDIAN with the help of graph

⇨ Draw less than and greater than cumulative frequency curves in one graph

⇨ From the point of intersection, draw a perpendicular to the x-axis to locate the value of median.

Note ; in the case of an inclusive series, convert it into exclusive

Problems

1. Obtain the value of median from the following

a. 391, 384, 407, 672, 522, 777, 753, 2488, 1490 ans 672

b. 22, 16, 18, 13, 15, 19, 17, 20, 23 ans 18

c.ans = 16

|Wage |Number of workers |

|10 |2 |

|12 |5 |

|14 |12 |

|16 |20 |

|18 |10 |

|20 |7 |

|22 |3 |

d. . ans =61

|X |f |

|58 |2 |

|59 |3 |

|60 |6 |

|61 |15 |

|62 |10 |

|63 |5 |

|64 |4 |

|65 |3 |

|66 |1 |

E ans =30

|Mid value |Frequency |

|10 |4 |

|20 |8 |

|30 |14 |

|40 |10 |

|50 |2 |

F ans = 35.5

|Marks less than |No. of students |

|10 |4 |

|20 |8 |

|30 |25 |

|40 |34 |

|50 |31 |

|60 |46 |

|70 |50 |

|80 |60 |

Quartiles

Quartiles divide a distribution into four equal parts.

There are three quartiles Q1, Q2, and Q3

Q1 is called the lower quartile. the number of values less than q1 is one fourth and more than q1 is three fourth of the total number of values in the set.

Q3 is called the upper quartile – three fourth of the total number of values in the set are less than q3 and one fourth values are more than Q3

Q2 is the median or second quartile, which divides the distribution into two equal parts.

Calculation of quartiles

CALCULATION OF Q1

1. Individual series

• Arrange the data in ascending or descending order

• Q1= size of (N+1) / 4 th item, (N is the number of items in the series)

2. Discrete series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find (N+1) /4, (N is the total frequency)

• Locate the total, which is equal to or greater than (N+1) /4 in the cumulative frequency column and determine the value of the variable corresponding to this to get the value of Q1

3. Continuous series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find (N) /4, (N is the total frequency) in the cumulative frequency column and determine the Q1 class

• Apply the following formula to find the value

Q1= L +( N/4 – C.f) / f x i

✓ L = lower limit of the quartile class

✓ N = total frequency

✓ Cf = cumulative frequency of the class above the Q1 class

✓ I = class width of theQ1 class

✓ F = is the frequency of the Q1class

CALCULATION OF Q3

1. Individual series

• Arrange the data in ascending or descending order

• Q3= size of 3(N+1) / 4 th item, (N is the number of items in the series)

2. Discrete series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find 3(N+1) /4, (N is the total frequency)

• Locate the total, which is equal to or greater than 3 (N+1) /4 in the cumulative frequency column and determine the value of the variable corresponding to this to get the value of median

3. Continuous series

• Arrange the data in ascending or descending order

• Find the cumulative frequencies

• Find 3(N) /4, (N is the total frequency) in the cumulative frequency column and determine the third quartile class

• Apply the following formula to find the value

• Q3 = L +(3 N/4 – C.f) / f x i

In the above formula L = lower limit of the Q3 class

N = total frequency

Cf = cumulative frequency of the class above the Q3 class

I = class width of the Q3 class

F = is the frequency of the Q3 class

(Since Q2 and median are the same, calculation is similar to median)

Numerals

1. Calculate lower quartile and upper quartile from the following

|Wages | 38 |

|30-35 |14 |

|35-40 |16 |

|40-45 |18 |

|45-50 |23 |

|50-55 |18 |

|55-60 |8 |

|60-65 |3 |

Ans Q1= 38.43 Q3 = 51.11

6. Find Q1, Q2, and Q3 from the following

| Marks below 10 20 30 40 50 60 70 80 |

| No. of students 15 35 60 84 96 127 198 250 |

Ans Q1 = 31.04 Q2 = 28.33 Q3 = 37.78

MODE

Arithmetic mean sometimes becomes unworthy due to the influence of extreme scores in the distribution. For example, in the series 2,2,3,4,10,12,20 and 45, the extreme scores detract the real value of the arithmetic mean

The median also fails to represent the typical central value, if the scores are divergent from each other, say as in series 10,30,100,150and 1000

In order to find a measure, which is free from the above defects, we look for the commonly occurring value, in a series called modal value. Thus, Mode is the value in a series, which occurs most frequently.

Calculation of mode in individual series.

In individual series mode can be calculated by just an inspection of the series. Find the

Value, which repeats maximum number of times in the series.

When there are two or more values repeating the same maximum number of times, we say mode is ill defined.

Calculation of mode in discrete series

There are two methods for calculating mode

Method of inspection –similar to individual series. However, where the mode is determined by just inspection , an error of judgment is possible in those cases where the difference between the maximum frequency and the next highest is very small or if there are two values with same frequency. In such cases, it is desirable to prepare a grouping table and analysis table.

Grouping method – grouping method consists of a grouping table and an analysis table.

Grouping table has six columns

Column 1 – frequencies are entered as such

Column2 - frequencies are grouped in two’s (sum of first two next two and so on)

Column 3 – frequencies are grouped in twos leaving the first one

Column 4 - frequencies are grouped in three

Column 5 – frequencies are grouped in three leaving the first one

Column 6 – frequencies are grouped in three leaving the first two

After preparing the grouping table, prepare an analysis table

The values against which frequencies are the highest are then entered by means of tally bar in relevant boxes

Calculation of mode in continuous series

Steps

By preparing the grouping table and analysis table or by inspection ascertain the modal class

Determine the value of mode by applying the formula

Mode = L + (D1/D1+D2) i

D1 = Difference between the frequency of the modal class and the frequency of the preceding class

D2 = Difference between the frequency of the modal class and the frequency of the succeeding class

L = lower limit of the modal class

I = the class width of the modal class

Location of mode graphically

• Draw the histogram of the given data

• Draw two lines diagonally in the middle of the modal class bar, starting from each upper corner of the bar to the upper corner of the adjacent bar.

• Draw a perpendicular line from the intersection of the two diagonals to the x – axis to get the value of the mode.

|For merits demerits of various measures of central tendency, refer the text book |

Calculate mode from the following

1.

|23 |

| workers 4 6 9 3 2 1 |

ans 75

4.

|Income |0-10 |10-20 |20-30 |30-40 |40-50 |

|families |28 |46 |54 |42 |30 |

Ans 24

5.

|Size |0-5 |5-10 |10-15 |15-20 |20-25 |25-30 |30-35 |

|frequency |1 |2 |10 |4 |10 |9 |2 |

An24.28

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