Location – Measurement of Central Tendency SS1 Further Mathematics Lesson Note

MEASUREMENT OF CENTRAL TENDENCY

Measures of central tendency: This is a measure of how the data are centrally placed. The three commonest measures of position, depending on the information required are the arithmetic mean, median and mode.

1. MEAN: It is the most widely used measure and is sometimes called the arithmetical average. The mean of the number x1, x2, x3, x4 …………………xn is given by:

   X = ∑x/n   where ∑x is the sum of all items.     n = number of items

When the data involves frequency; mean = ∑fx/∑f

Examples:

1. Calculate the mean of the numbers 15, 17, 19, 21, 23, 25, 27, 29.

Solution:

Mean (x) = (15 + 17 + 19 + 21 + 23 + 25 + 27 + 29) ÷ 8  = 176/8 = 22

2. The table shows the number of suitcases possessed by a group of travellers.

Calculate the mean to the nearest whole number.

Solution:                                                            

    = Mean ( x ) = ∑ fx/∑ f    =  84/30 =   2.8  = 3   

Total            30                  84

2. Mode: The mode of a distribution is the value of the variable which occurs most often in the distribution. It is also possible for a distribution to have more than one mode if there is more than one item having the highest frequency.

Example:

Find the mode of the data 5, 4, 8, 9, 6, 8, 9, 3, 8. 

The mode is 8 (it appears 3 times more than others)

3. Median: This is the middle value of a set of data when arranged in ascending or descending order.

Example:

Find the median of these numbers: (1). 35, 28, 42, 28, 56, 70, 35      

(2) 18, 20, 25, 30, 22, 25, 28, 15

Solution:

Rearranging the numbers: 70, 56, 42, [35] 35, 28, 28. The median is 35

15, 18, 20, [22, 25], 25, 28, 30.   Median = 22 + 25 =   47    = 23.5

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MEAN, MEDIAN AND MODE OF GROUPED DATA

Mean: The arithmetic mean of grouped frequency distribution can be obtained using:

1. Class Mark Method:

X  =      where x is the midpoint of the class interval.

2. Assumed Mean Method: It is also called working mean method.    

X  =  A + (∑ fd/∑f)    

Where, d = x – A,   x = class mark and A = assumed mean.

Example: The numbers of matches in 100 boxes are counted and the results are shown in the table below:                                                                                     

Calculate the mean (i) using the class mark    (ii) assumed mean method given that the assumed mean is 30.5.

Solution:

1. Class Mark Method: X  =     

=  3214/100   = 32. 14 = 32 matches per box (nearest whole no)                                                 

1. Assumed Mean Method: 

X  =  A + (∑ fd/∑f)    

= 30. 5 + (164/100) =30.5 + 1.64

= 32.14 = 32 matches per box (nearest whole number)

Mode

The mode of a grouped frequency distribution can be determined geometrically and by interpolation method.

1. Mode from Histogram: The highest bar is the modal class and the mode can be determined by drawing a straight line from the right top corner of the bar to the right top corner of the adjacent bar on the left. Draw another line from the left top corner to the bar of the modal class to the left top corner of the adjacent bar on the right.
2. MODE FROM INTERPOLATION: The mode can be obtained using the formula.                                    

Mode =  Lm  +        ∆1   {C }

                            ∆1 + ∆2                                                                                Where Lm = lower class boundary of the modal class.

∆1  = difference between the frequency of the modal class and the class before it.

∆2  = difference between the frequency of the modal class and the class after it.

C   = class width of the modal class.

MEDIAN FROM INTERPOLATION FORMULA

Median = L1 +   N/2 – [ cfm/fm] C

Where, L1 = lower class boundary of the median class.

Cfm = cumulative frequency of the class before the median class.

Fm = frequency of the median class.

C   = class width of the median class and N   = Total frequency