Statistics: Measurement of central tendency

Measurement of central tendency

• Mean, Median and Mode of ungrouped
• Mean, Median and Mode of grouped data

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 the mode.

MEAN, MEDIAN AND MODE OF GROUPED DATA

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

 Class Mark Method:

            X  = 

    where x is the midpoint of the class interval.

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:

Number of matches	25    –   28	29    –    32	33   –    36	37      –     40
Number of boxes	18	34	37	11

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

Solution:

Class interval	F	X	FX	d = x – A	Fd
25     –     28	18	26.5	477	4	72
29     –     32	34	30.5	1037	0	0
33     –     36	37	34.5	1276.5	4	148
37     –     40	11	38.5	423.5	8	88
Total	100		3214		164

• Class Mark Method: X  =     =  3214/100   = 32. 14 = 32 matches per box (nearest whole no)                                                
• Assumed Mean Method: X  =  A + (∑ fd/∑f)   

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

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

Evaluation:

Calculate the mean shoe sizes of the number of shoes represented in the table below using (i) class mark   (ii) assumed mean method given that the assumed mean is 42.

Shoe sizes	30    –      34	35      –    39	40     –     44	45      –    49	50      –   54
No of Men	10	12	8	15	5

Mode

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

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.

Reading Assignment: Further Mathematics Project Book 1(New third edition), pg 328, Exercise18, No 15 -20

Weekend Assignment         

Marks	3	4	5	6	7	8
Frequency	5	x – 1	x	9	4	1

If the mean is 5, calculate the (a) value of x     (b) mode      (c) median of the distribution.

2.         The table gives the frequency distribution of a random sample of 250 steel bolts according to their head diameter, measured to the nearest 0.01mm.

Diameter (mm)	23.06 –23.10	23.11 – 23.15	23.16 –23.20	23.21 –23.25	23.26-23.30	23.31 –23.35	23.36-23.40	23.41-23.45	23.46-23.50
No of bolts	10	20	28	36	52	38	32	21	13

1. State the median class and calculate the median using interpolation method.
2. Draw the histogram and use it to estimate the mode.
3. Calculate the mean value using a working mean of 23.28mm.
4. The table gives the frequency distribution of marks obtained by a group of students in a test.