Measures of Dispersion

Dr Anshul Singh Thapa

MEASURES OF DISPERSION FROM�AVERAGE

• Recall that dispersion was defined as the extent to which values differ from their average. Range and Quartile Deviation are not useful in measuring, how far the values are, from their average. Yet, by calculating the spread of values, they do give a good idea about the dispersion. Two measures which are based upon deviation of the values from their average are Mean Deviation and Standard Deviation.
• Since the average is a central value, some deviations are positive and some are negative. If these are added as they are, the sum will not reveal anything. In fact, the sum of deviations from Arithmetic Mean is always zero.
• Mean Deviation tries to overcome this problem by ignoring the signs of deviations, i.e., it considers all deviations positive. For standard deviation, the deviations are first squared and averaged and then square root of the average is found. We shall now discuss them separately in detail.

Mean Deviation

• Mean Deviation is the arithmetic average of the deviations of all the values taken from some average value (mean, median, mode) of the series, ignoring signs (+ or -) of the deviations.
• Clark and Schakde “Mean deviation is the arithmetic average of deviations of all the values taken from a statistical average (mean, median or mode) of series. In taking deviation of values, Algebraic signs + and - are not taken into consideration, that is negative deviations are also treated as positive deviation.

MEAN DEVIATION

UNGROUPED DATA

DISCRETE SERIES

DIRECT METHOD

ASSUMED MEAN METHOD

FROM MEDIAN

GROUPED DATA

FROM ARITHMETIC MEAN

CONTINUOUS SERIES

DIRECT METHOD

SHORT CUT METHOD

FROM ARITHMETIC MEAN

SHORT CUT METHOD

SHORT CUT METHOD

FROM MEDIAN

FROM MEDIAN

FROM ARITHMETIC MEAN

DIRECT METHOD

DIRECT METHOD

MEAN DEVIATION

UNGROUPED DATA

DISCRETE SERIES

FROM MEDIAN

GROUPED DATA

FROM ARITHMETIC MEAN

CONTINUOUS SERIES

FROM ARITHMETIC MEAN

FROM MEDIAN

FROM MEDIAN

FROM ARITHMETIC MEAN

ΣI dx I

n

ΣI dx I

n

ΣI dx I

n

ΣI dx I

n

ΣI dx I

n

ΣI dx I

n

CALCULATION OF MEAN DEVIATION FOR UNGROUPED DATA

Calculation of Mean Deviation from Arithmetic Mean for ungrouped data – Direct Method

Steps:

• The Arithmetic Mean of the values is calculated
• Difference between each value and the Arithmetic Mean is calculated. All differences are considered positive. These are denoted as |d|
• The Arithmetic Mean of these differences (called deviations) is the Mean Deviation.

Example

Example

Example

Example

Example

Example

• The M D_x = Σ|dx| = 12 = 2.4

N 5

Calculation of Mean Deviation from Arithmetic Mean for ungrouped data – Assumed Mean Method

• Mean Deviation can also be calculated by calculating deviations from an assumed mean. This method is adopted especially when the actual mean is a fractional number. (Take care that the assumed mean is close to the true mean). For the values in example 3, suppose value 7 is taken as assumed mean, M.D. can be calculated as under:

Example

• In such cases, the following formula is used:

MD_x = Σ|dx| + (X – Ax) (Σf_B – Σf_A)

n

• Where Σ|dx| is the sum of absolute deviations taken from the assumed mean.
• x is the actual mean.
• Ax is the assumed mean used to calculate deviations.
• Σf_B is the number of values below the actual mean including the actual mean.
• Σf_A is the number of values above the actual mean.
• Substituting the values in the above formula:

MD_x = 11 + (6 – 7) (3 – 2) = 12 = 2.4

5 5

Mean Deviation from median for ungrouped data

Direct Method

• M.D. from the Median can be calculated as follows:
• Calculate the median which is 4400 in this case.
• Calculate the absolute deviations from median, denote them as |d|.
• Find the average of these absolute deviations. It is the Mean Deviation.

Mean Deviation from median for ungrouped data

Median = Size of N + 1 th item = 5+ 1 = 3^rd item

2 2

Size of the third item is 4400

M.D._median = |dx| = 1200 = 240

n 5

​

Mean Deviation from median for ungrouped data

Median = Size of N + 1 th item = 5+ 1 = 3^rd item

2 2

Size of the third item is 4400

Mean Deviation from median for ungrouped data

Median = Size of N + 1 th item = 5+ 1 = 3^rd item

2 2

Size of the third item is 4400

Mean Deviation from median for ungrouped data

Median = Size of N + 1 th item = 5+ 1 = 3^rd item

2 2

Size of the third item is 4400

Mean Deviation from median for ungrouped data

Median = Size of N + 1 th item = 5+ 1 = 3^rd item

2 2

Size of the third item is 4400

M.D._median = |dx| = 1200 = 240

n 5

​

Mean Deviation from median for ungrouped data

Short-cut method

• To calculate Mean Deviation by short cut method, a value (A) is used to calculate the deviations and the following formula is applied:

MD_median = Σ|dx| + (Median – A) (Σf_B – Σf_A)

n

• Where, A = the constant from which deviations are calculated. (Other notations are the same as given in the assumed mean method).

​

​

Mean Deviation from mean for discrete series

• Actual Mean = Σfm = 576 = 12

Σf 48

​

• Actual Mean = Σfm = 576 = 12

Σf 48

​

• Actual Mean = Σfm = 576 = 12

Σf 48

​

• Actual Mean = Σfm = 576 = 12

Σf 48

​

• Actual Mean = Σfm = 576 = 12

Σf 48

• M.D._x = Σf|d| = 36 = 0.75

Σf 48

​

Mean Deviation from mean for discrete series

Mean Deviation _Mean = Σf|d| = 338.59 = 1.74

Σf 195

Mean Deviation from median for discrete series

• Median = size of (N+1)/2 th item
• Median = (48 + 1) = 49 = 24.5^th item

2

Size of 5^th item lies in 33 cumulative frequecy therefore the median is 12

​

• Median = size of (N+1)/2 th item
• Median = (48 + 1) = 49 = 24.5^th item

2

Size of 5^th item lies in 33cumulative frequecy therefore the median is 12

​

• Median = size of (N+1)/2 th item
• Median = (48 + 1) = 49 = 24.5^th item

2

Size of 5^th item lies in 33cmlativefrequecy therefore the median is 12

​

• Median = size of (N+1)/2 th item
• Median = (48 + 1) = 49 = 24.5^th item

2

Size of 5^th item lies in 33cmlativefrequecy therefore the median is 12

• Mean Deviation _median = Σf|d| = 36 = 0.75

Σf 48

​

Mean Deviation from Mean for�Continuous distribution

Steps:

• Calculate the mean of the distribution.
• Calculate the absolute deviations |d| of the class midpoints from the mean.
• Multiply each |d| value with its corresponding frequency to get f|d| values. Sum them up to get Σf|d|.
• Apply the formula

​

Example

• Actual Mean = Σfm = 1620 = 40.5

Σf 40

• M.D._x = Σf|d| = 519 = 12.97

Σf 40

• Actual Mean = Σfm = 1620 = 40.5

Σf 40

• M.D._x = Σf|d| = 519 = 12.97

Σf 40

• Actual Mean = Σfm = 1620 = 40.5

Σf 40

• M.D._x = Σf|d| = 519 = 12.97

Σf 40

• Actual Mean = Σfm = 1620 = 40.5

Σf 40

• M.D._x = Σf|d| = 519 = 12.97

Σf 40

• Actual Mean = Σfm = 1620 = 40.5

Σf 40

• M.D._x = Σf|d| = 519 = 12.97

Σf 40

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Median Class: N/2 = 50/2 = 25

Median = L + (N/2 - c.f.) × i

f

Median = 40 + (25 - 15) x 20 = 50

20

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Mean Deviation from median for�Continuous distribution

• The procedure to calculate Mean Deviation from the median is the same as it is in case of M.D. from Mean, except that deviations are to be taken from the median as given below:

Median = L + (N/2 - c.f.) × i

f

Median = 40 + (25 - 15) x 20 = 50

20

Mean Deviation _median = Σf|d| = 665

Σf 50