Question

Using the frequency distribution table given below, draw 'less than ogive'. Then from the ogive, find interquartile range.

$\begin{array}{cc}Classinterval& Frequency\\ 10-20& 2\\ 20-30& 4\\ 30-40& 6\\ 40-50& 8\\ 50-60& 10\end{array}$

Answer 1

We prepare the cumulative frequency table as given below:
$\begin{array}{ccc}Classinterval& Frequency& Cumulative\\ & & frequency\\ 10-20& 2& 2\\ 20-30& 4& 6\\ 30-40& 6& 12\\ 40-50& 8& 20\\ 50-60& 10& 30\\ & & \\ hline& & \end{array}$
We need to plot the points (10, 0), (20, 2), (30, 6), (40, 12), (50, 20) and (60, 30). Plotting these points on a graph paper and joining by a freehand curve, we get the cumulative frequency curve (a 'less than ogive') as given below:

Finding the lower quartile and upper quartile to find the inter-quartile range

i) To find the lower quartile $Q_1$, we locate $\frac{n+1}{4}$, i.e., $\frac{31}{4}$ =7.5 on y-axis and proceed horizontally to meet the ogive. From this point, draw a perpendicular to meet the x-axis.
$\therefore$ From figure $Q_1=32.5$

ii) To find the upper quartile $Q_3$, we proceed similarly as in (i) taking
$\frac{3(n+1)}{4}=\left(\frac{3\times 7.75}{4}\right)=23.25$ on y-axis
Then, $Q_3=53.5$.

iii) Interquartile range $Q_3-Q_1$= 53.5 - 32.5 = 21