Question

From the following data, draw the two types of cumulative frequency curves and determine the median:

Height (in cm)	Frequency
140-144	3
144-148	9
148-152	24
152-156	31
156-160	42
160-164	64
164-168	75
168-172	82
172-176	86
176-180	34

Answer 1

(i) Less than series:

Marks	No. of students
Less than 144	3
Less than 148	12
Less than 152	36
Less than 156	67
Less than 160	109
Less than 164	173
Less than 168	248
Less than 172	330
Less than 176	416
Less than 180	450

We plot the points A(144,3), B(148,12), C(152,36), D(156,67), E(160,109) F(164,173), G(168,248), H(172,330), I(176,416) and J(180,450). Join AB, BC, CD, DE, EF, FG, GH, HI, IJ and JA with a free hand to get the curve representing the ‘less than type’ series.

(ii) More than series:

Marks	No. of students
More than 140	450
More than 144	447
More than 148	438
More than 152	414
More than 156	383
More than 160	341
More than 164	277
More than 168	202
More than 172	120
More than 176	34

Now on the same graph paper, we plot the points $A_1$(140,450), $B_1$(144,447), $C_1$(148,438), $D_1$(152,414), $E_1$(156,383), $F_1$(160,341), $G_1$(164,277), $H_1$(168,202), $I_1$(172,120) and $J_1($176,34).
Join $A_1{B}_1,B_1{C}_1,C_1{D}_1,D_1{E}_1,E_1{F}_1,F_1{G}_1,G_1{H}_1,H_1{I}_1\mathrm{and}{I}_1{J}_1$ with a free hand to get the ‘more than type’ series.

The two curves intersect at point L. Draw $\mathrm{LM}\perp \mathrm{OX}$ cutting the $x-axis$at M.
Clearly, M = 166 cm
Hence, Median = 166 cm