Question

A Grade 7 teacher asked his students to pick a two-digit number below 30. He collected the data and represented it as shown below.

$20$

$18$

$19$

$28$

$24$

$26$

$22$

$21$

$23$

$27$

$20$

$25$

$26$

$23$

$28$

$24$

Which among the following box and whisker plots resembles the given data set?

Answer 1

After arranging the given data set in ascending order, we get:

$18$

$19$

$20$

$20$

$21$

$22$

$23$

$23$

$24$

$24$

$25$

$26$

$26$

$27$

$28$

$28$

From the above data set, we get:
Lower extreme = $18$
Upper extreme = $28$
Middle quartile = Median = Mean of $23$ and $24$
$\Rightarrow$ Middle quartile = $\frac{23+24}{2}=23.5$

$\bullet$ To calculate the lower quartile, we have to consider the left part of middle quartile $\left(23.5\right)$ data set.

$18$

$19$

$20$

$20$

$21$

$22$

$23$

$23$

Lower quartile = Median of the data set above = Mean of $20$ and $21$
$\Rightarrow$ Lower quartile = $\frac{20+21}{2}=20.5$

$\bullet$ To calculate the upper quartile, we have to consider the right part of middle quartile $\left(23.5\right)$ data set.

$24$

$24$

$25$

$26$

$26$

$27$

$28$

$28$

Upper quartile = Median of the above data set = Mean of $26$ and $26$
$\Rightarrow$ Upper quartile = $\frac{26+26}{2}=26$

To draw the box and whisker plot for the given data set, we follow the following steps:

1. Mark the lower and upper extremes as $18$ and $28$, respectively (dots in cyan).
2. Mark the middle quartile $23.5$ (dot in green).
3. Mark lower and upper quartiles as $20.5$ and $26$, respectively (dots in orange).
4. Draw the box covering the lower and upper quartiles, i.e., between $20.5$ and $26$.
5. Extend the whiskers from the side of the box upto the lower and upper extremes, i.e., upto $18$ and $28$.