Name: Variance and Standard Deviation for Ungrouped Distribution
Uploaded: 2019-07-19
Duration: 6 min 21 s
Description: In this video, let us learn to find the variance and standard deviation of ungrouped data. We will also learn the formula to find the discrete frequency dist...

Question

The following stem-and-leaf diagram shows the marks out of $50$ obtained by $15$ students in a Science test.

Find the range and interquartile range of the marks.

Accepted Answer

In a stem-and-leaf diagram: The stem represents the category figure, the leaves represent the final digit(s) of each data point and the key tells us how to read the values

Let number of data values be $N$ .

$\Rightarrow N = 15$

We know that, lower quartile $(Q_{1})$ is the median of the lower $50 %$ of the data.

Formula to find lower quartile is $Q_{1} = {(\frac{N + 1}{4})}^{th}$ data value, where $N$ is number of data values.

$\Rightarrow$ Lower quartile, $Q_{1} = {(\frac{15 + 1}{4})}^{th} = 4^{th}$ data value.

The $4^{th}$ data value in the stem-and- leaf plot is $28$ .

Hence, the lower quartile of the given data is $Q_{1} = 28$ marks.

Now, upper quartile $(Q_{3})$ is the median of the upper $50 %$ of the data.

Formula to find upper quartile is $Q_{3} = {(\frac{3 (N + 1)}{4})}^{th}$ data value, where $N$ is number of data values.

$\Rightarrow$ Upper quartile $Q_{3} = {(\frac{3 (15 + 1)}{4})}^{th} = 12^{th}$ data value.

The $12^{th}$ data value in the stem-and- leaf plot is $46$ .

Hence, the upper quartile of the given data is $Q_{3} = 46$ marks.

Interquartile range is $IQR = Q_{3} - Q_{1}$

Therefore, $IQR = 46 - 28 = 18$ marks.

Now, Range $=$ Difference between the highest value and lowest value.

The highest number in the stem-and-leaf plot is the last stem and the last leaf on the chart and the lowest number is the first stem and the first leaf on the chart.

Here, the highest value is $50$ and the lowest value is $9$ .

Therefore, range $= 50 - 9 = 41$ .

Deviation from $90 ° (d)$	No. brackets $(f)$
$d < - 1.5$	$0$
$- 1.5 ⩽ d < - 1.0$	$24$
$- 1.0 ⩽ d < - 0.5$	$46$
$- 0.5 ⩽ d < 0.0$	$61$
$0.0 ⩽ d < 0.5$	$34$
$0.5 ⩽ d < 1.0$	$34$
$1.0 ⩽ d < 1.5$	$20$
$1.5 ⩽ d < 2.5$	$17$

$x$	$< 0$	$< 10$	$< 15$	$< 25$	$< 30$	$< 40$
$c f$	$0$	$12$	$30$	$90$	$102$	$120$

Biology	Chemistry	Physics
$21 33 45$	$41 53 65$	$51 63 75$

Biology	Chemistry	Physics
$21 33 45$	$41 53 65$	$51 63 75$

Time spent (min)	$20 - 30$	$30 - 40$	$40 - 50$	$60 - 80$
No. girls $(f)$	$6$	$14$	$7$	$3$
No. boys $(f)$	$15$	$11$	$7$	$7$

The following stem-and-leaf diagram shows the marks out of $50$ obtained by $15$ students in a Science test.

Find the range and interquartile range of the marks.

Important Questions on Measures of Variation

Learn and Prepare for any exam you want !

Time spent (min)	$20 -$	$30 -$	$40 -$	$60 - 80$
No. girls $(f)$	$6$	$14$	$7$	$3$
No. boys $(f)$	$15$	$11$	$7$	$7$

The following stem-and-leaf diagram shows the marks out of 50 obtained by 15 students in a Science test. Find the range and interquartile range of the marks.

Important Questions on Measures of Variation

Learn and Prepare for any exam you want !

The following stem-and-leaf diagram shows the marks out of $50$ obtained by $15$ students in a Science test.

Find the range and interquartile range of the marks.