Question

Grouped, continuous data for the mass, $w kg$ , of a group of adults is given in the table below.

$\begin{array}{lcccccccc} Mass & 40 < w \leq 50 & 50 < w \leq 60 & 60 < w \leq 70 & 70 < w \leq 80 & 80 < w \leq 90 & 90 < w \leq 100 & 100 < w \leq 110 & 110 < w \leq 120 \\ Frequency & 5 & 15 & 25 & 30 & 50 & 35 & 25 & 15 \end{array}$

Hence, estimate values for the lower quartile. Draw lines on your graph to indicate how you obtained these values.

Accepted Answer

Given data:

$\begin{array}{lcccccccc} Mass & 40 < w \leq 50 & 50 < w \leq 60 & 60 < w \leq 70 & 70 < w \leq 80 & 80 < w \leq 90 & 90 < w \leq 100 & 100 < w \leq 110 & 110 < w \leq 120 \\ Frequency & 5 & 15 & 25 & 30 & 50 & 35 & 25 & 15 \end{array}$

The cumulative frequency is the sum of the frequencies for every class, up to and including the current one. A table which displays the manner in which cumulative frequencies are distributed over various classes is called a cumulative frequency distribution or cumulative frequency table.

Cumulative frequency distribution table for the given data is

$\begin{array}{lcccccccc} Mass & w \leq 50 & w \leq 60 & w \leq 70 & w \leq 80 & w \leq 90 & w \leq 100 & w \leq 110 & w \leq 120 \\ \begin{array}{l} Cumulative frequency \end{array} & 5 & 5 + 15 = 20 & 5 + 15 + 25 = 45 & 5 + 15 + 25 + 30 = 75 & 5 + 15 + 25 + 30 + 50 = 125 & 5 + 15 + 25 + 30 + 50 + 35 = 160 & 5 + 15 + 25 + 30 + 50 + 35 + 25 = 185 & 5 + 15 + 25 + 30 + 50 + 35 + 25 + 15 = 200 \end{array}$

Steps to draw cumulative frequency curve:

(i) Draw and mark the horizontal and vertical axes.

(ii) Take the cumulative frequencies along the $y$ -axis (vertical axis) and the end points of the class limits on the $x$ -axis (horizontal axis).

(iii) Against each end points of limit, plot the corresponding cumulative frequencies.

(iv) Connect the points with a continuous curve.

Cumulative frequency curve of the given data is

To find the lower quartile, we draw a horizontal line across from the frequency axis to the curve at the ${(\frac{n}{4})}^{th}$ value and then down to the $x$ -axis.

Here, $n = 200$

So, ${(\frac{n}{4})}^{th}$ value is $50^{th}$ value.

Draw a line across from $50$ on the vertical axis( $y$ -axis) until it hits the curve and then draw a line vertically down until it touches the horizontal axis( $x$ -axis). This is the lower quartile $(Q_{1})$ .

Thus, the lower quartile of the given data is $73$ .

Grade	$1$	$2$	$3$	$4$	$5$	$6$	$7$
Frequency	$4$	$8$	$16$	$20$	$16$	$8$	$4$

Grade	$1$	$2$	$3$	$4$	$5$	$6$	$7$
Frequency	$4$	$8$	$16$	$20$	$16$	$8$	$4$

Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Representing Data: Statistics for Univariate Data, Exercise 43: Chapter review

Natasha Awada Mathematics Solutions for Exercise - Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Representing Data: Statistics for Univariate Data, Exercise 43: Chapter review

Questions from Natasha Awada, Paul La Rondie, Laurie Buchanan and, Jill Stevens Solutions for Chapter: Representing Data: Statistics for Univariate Data, Exercise 43: Chapter review with Hints & Solutions

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