Measures of Dispersion-Ungrouped data

Lecture # 5

To find the variance, standard deviations, standard error, mean deviation, range and skewness.

Definitions:

Sample variance =∑(x-x )2 /n-1= S2 = (n∑x2 –(∑x)2)/ n(n-1)

Where

X = sample arithmetic mean

n = sample size

(2) Standard deviation = S = square root of the variance .

(3) Standard error =  standard deviation / square root of n

(4) Mean deviation = ∑ │x-x │/ n

(5) Range = highest value – lowest value

(6) Coefficient of variation =

(standard deviation/mean) . 100

(7) Skewness = 3( mean – median ) / standard deviation

EXAMPLE :

Given the data  9  ,7  ,11  ,10  ,13  ,and 7

Calculate : mean , median , variance , standard deviation , mean deviation , range , coefficient of variation and skewness .

Using the previous definitions :

Mean = ∑xi / n = 57 / 6 = 9.5

For the median , first we find it’s position  ;  n+1 / 2 =3.5  ,  then the  median is the mean of the values of the 3rd and the 4th items (after arranging in order ) .

median =(9+10) /2= 9.5

To calculate the variance construct the following table :

X        X – X          (X – X )2

______________________________

9        – 0.5               0.25

7         -2.5               6.25

11         1.5                2.25

10         0.5                0.25

13         3.5              12.25

7        -2.5                6.25

______________________________

Total         0.0              27.50

The variance = S2 = 27.50 / 5 = 5.5

The standard deviation =√ S2 = √ 5.5 = 2.3

Standard error = S/√n =2.3 / √6 = 0.93

Mean deviation = 11 / 6 = 1.83

Range = 13 – 7 = 6

Coefficient of variation = (2.3 / 9.5 ) x 100 = 24.21 %

Skewness = 3( 9.5 – 9.5 ) / 2.3 = 0

Discussion :

We notice in this example that the mean and the median are equal so the distribution is approximately uniform.

For symmetric distributions :

Mean = median = mode

For positively skewed distribution:

Mean > median > mode

For negatively skewed distribution:

Mean < median < mode

EXPERIMENT(1) :

Following are the number of patients visited a clinic over each of the last 30 days .

83     64     84     76     84     54     75     59     70     61

63     80     84     73     68     52     65     90     52     77

95     36     78     61     59     84     95     47     87     60

(1) Determine the mean number of patients visited the clinic per day ?

(2) What is the median using number?

(3) What is the modal using time?

(4) Determine the standard deviation for the using time.

(5) Determine the coefficient of variation and the coefficient of skewness.

(6) Draw a box plot.

(7) Write a brief report summarizing the results.

EXPERIMENT(2) :

Following are the number of times an ATM machine was used over each of the last 30 days.

83     64     84     76     84     54     75     59     70     61

63     80     84     73     68     52     65     90     52     77

95      36     78     61     59     84     95     47     87     60

(1) Determine the number of times the machine was used per day ?

(2) What is the median using number ?

(3) What is the modal using time ?

(4) Determine the standard deviation for the using time, the coefficient of variation and the coefficient of skewness.

(5) Draw a box plot .

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