mode grouped

5. Modę of a grouped frequency distribution by formuła    ~

An estimate of tbe modę for a grouped freąuency distribution can be obtained u Bing- the following procedurę,. STEP 1    - Determine the modal class (that class which has the largest freąuency).

STEP 2

Calculate D₁ — difference between the largest freąuency and the freąuency immediately pręt

it    , ,_;V

STEP 3 STEP 4

Calculate D _z — difference between the largest freąuency and the freąuency immediately followin0 Use the following interpolation formula:

' -'y

8 Modę and other measures of location

Interpolation formula for the modę

r ^di

Modę = L +

.C

Dy +D₂ _

L = lower bound of modal class C = modal class width and:    Dy, D₂ are as described above in STEPS 2 and 3.

where:	L
	C
and:	Dy, Dty

-

Example 2 demonstrates the use of this procedurę.

6. Example 2 (Estimation of the modę of a freąuency distribution using the interpolation formula)

Question

Estimate the modę of the following distribution of ages.

r

Age (years)	20-25	25-30	30-35	35-40	40-45	45-50
Number of employees	2	14	29	43	33	9

v.-:

Answer

The table below shows the standard layout of the data, with the steps in the procedurę clearły specified.

STĘPI

Modal class

Age (years)	Number of employees
20 and under 25	2
25 and under 30	14 ^
i 30 and under 35	29jc
35 and under 40	43^_
40 and under 45	33
45 and under 50	9

STEPS 2 AND 3

D = 43-29 = 14

D = 43-33 = 10

2

STEP 4

The lower class bound of the modal class, L - 35 The class width of the modal class, C = 5 (from 35 to 40)

D,

Thus:

modę = L +

= 35 +

Dy+D₂_

14

.C

.5

i.e.

.14 + 10. modę = 37.92 years (2D)

7. Graphical estimation of the modę

The graphical eąuivalent of the above interpolation formula is to construct three histogram bars, repres the class with the highest freąuency and the ones on either side of it, and to draw two lines, as shown in

1 ^rP'U ___l ____j ii______i_____—------__j—    _______ u___