Measures of Central Tendency: Mean & weighted mean, Mode, Median, Midrange.                                                                                       

Arithmetic Mean:                                                                                   

Ungrouped data: Mean = Ʃ x / n                                                                   

Grouped data:      Mean = Ʃ f X / n where f refers to the frequency and x refers to the observations or to midpoint classes.                            

Ungrouped data Mean  = Ʃ x  / n                                                                

Example: The ages of 5 patients are : 20 , 45 , 30 , 55 & 70 years , so the mean age of the group will be : ( 20 + 45 + 30 + 55 + 70 ) / 5  = 220 / 5 = 44 years.

Grouped data Mean ( X )  =  Ʃ f X / n  where f refers to the frequency and X refers to observations and n refers to the number.                           

Example: 40 individuals, 5 of which having 60 KG weight, 20 having 65 KG weight & 15 having 70 KG weight, so the mean weight will be            (

5 x 60 ) + ( 20 x 65 ) + ( 15 x 70 ) / 40 =  66.25 KG

Grouped data Mean ( X )  =  Ʃ f X / n  where f refers to the frequency and X refers to midpoint classes and n refers to the number

Simple frequency Height in cms
6159 –
15162 –
31165 –
22168 –
20171 –
4174 –
2177 +

Looking to the previous table, n = 6 +15+31+22+20+4+2 = 100 and f is the frequency which is 6 , 15 , 31 , 22 , 20 , 4 and 2                                      

There are 7 classes of data and the X which is the midpoint class is required to be obtained.                                                                                      

The midpoint class = the mean of 2 adjacent classes.                                   

So the 1st midpoint class = 159 +162 / 2 = 160.5, by knowing this 1st class, we can obtain the other classes by simply adding 3 to each class (3 is the class length) so the other midpoint classes will be 163.5, 166.5, 169.5, 172.5, 175.5 and 178.5

fMidpoint class
6160.5
15163.5
31166.5
22169.5
20172.5
4175.5
2178.5

By using the equation Mean = Ʃ f X / n, the mean will be                         

{(160.5 x 6) + (163.5 x 15) + (166.5 x 31) + (169.5 x 22) + (172.5 x 20) + (175.5 x4) + (178.5 x 2)} / 100 = 168.15 Cms                                                   

Weighted Mean: Is the mean of the means

It equals X1 x n1 + X2 x n2 + X3 x n3 + ……………….. / n1 + n2 + n3+ 

Weighted mean can be used as a rough estimate of the universe mean. If you don’t know the universe mean, obtain the mean from different surveys and calculate the weighted mean.                                               

Mode: is the most frequent observation.                                                        

Data may have no mode, may be uni – modal, bi- modal or poly- modal. Example: 1, 2, 3, 4, 5  is a no modal set of data while 1, 2, 2, 3, 4, 5  is     uni – modal and 1, 1, 2, 2, 3, 4, 5 is bi- modal  while 1, 1, 2, 2, 3, 3, 4, 5 is poly- modal.                                                                                                 

Median: is the number that bisects the observations into equal values. To obtain the median , arrange the values by order from the highest to lowest or from the lowest to highest then the order of the median value will be :

( n + 1 ) / 2  If the data are odd number. ( n / 2 ) & ( n/ 2 ) + 1  If the data are even number.

Example(1): Estimate the median for the following set of da12, 17, 9, 18, 14, 22, 26                                                                                         

This set of data is odd number (7 observations)                                            

The data are to be arranged: 9, 12, 14, 17, 18, 22, 26                                  

Then the order of the median will be (n + 1) / 2 = (7 + 1) / 2 = 4               

So the median for these data is the 4th number which is 17            

Example(2) : Estimate the median for the following set of data:              

14, 6, 8, 12, 10, 24                                                                                                

This set of data is even number (6 observations)                                           

The data are to be arranged: 6, 8, 10, 12, 14, 24                                            

hen the order of the median will be :  (n /2) & (n /2) + 1 = (6/2) & ( 6/2 ) + 1 = 3 & 4                                                 

So the median values for these data are the 3rd & 4th values, 10 &12       and the median = (10+12 ) / 2 = 11                                                          

Midrange: is the highest plus the lowest values divided by 2.              

Example: Range of normal Hb is 10 → 16 gm                                           

Midrange = ( 16 + 10 ) / 2 = 13

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