Statistics III

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In today’s class, we will be talking more about statistics. Enjoy the class!

Statistics III

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 the 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.

Example:

The table gives the distribution of ages of students in an institution.

Ages (year)	16    –      18	19      –    21	22     –     24	25      –    27	28      –   30
No of Students	18	30	35	24	13

Draw a histogram and use your histogram to estimate the mode to the nearest whole number.

Solution

Class Interval (Ages)	F	Class Boundary
16      –     18	18	15.5    –   18. 5
19     –      21	30	18.5    –   21.5
22     –      24	35	21.5    –   24.5
25     –      27	24	24.5    –  27.5
28     –      30	13	27.5   –   30.5

Example: Using the table given in the example above:

Modal class = 22 – 24,   ∆₁ = 35 – 30 = 5,  ∆₂ = 35 – 24 = 11,   C = 3,   L_m = 21.5

Mode = 21.5   +     5            3

5 + 11

= 21.5 + (15/16)   = 21.5 + 0.9375

= 22.44, approximately 22 yrs.

MEDIAN OF GROUPED DATA:

The median of grouped data can be determined from a cumulative frequency curve and the interpolation formula.

Median from Cumulative Frequency Curve: The cumulative frequency curve can be used to determine the median.

EXAMPLE: The table below shows the masses of 50 students in a secondary school

Masses (kg)	10    –   14	15    –   19	20  –     24	25    –   29	30  –  34	35 –  39	40    –     44
Frequency	3	7	9	5	11	6	9

1. Prepare a cumulative frequency table for the data.
2. Draw the ogive and use your graph to find the median.

GENERAL EVALUATION

The table below gives the distribution of masses (kg) of 40 people

Masses (kg)	1 – 5	6 – 10	11 -15	16    –   20	21     – 25	26    – 30	31     – 35	36    –   40
Frequency	9	20	32	42	35	22	15	5

1. State the modal class of the distribution and find the mode.
2. Draw a cumulative frequency curve to illustrate the distribution.
3. Use the curve in ‘2’ to estimate the median.
4. Calculate the mean of the distribution.

 

In our next class, we will be talking about Probability.  We hope you enjoyed the class.

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