Calculate the quartile deviation for the given grouped data.

Question

ScanSolve AI Tutor · Accepted Answer

To calculate the quartile deviation for the given grouped data, we first need to find the first quartile ( $Q_1$ ) and the third quartile ( $Q_3$ ).

The formula for the $k^{th}$ quartile ( $Q_k$ ) for grouped data is:

Q_k = L + \left(\frac{\frac{kn}{4} - cfb}{f}\right)w

Where:

$L$ = Lower boundary of the $k^{th}$ quartile class
$n$ = Total frequency ( $\sum f$ )
$cfb$ = Cumulative frequency of the class preceding the $k^{th}$ quartile class
$f$ = Frequency of the $k^{th}$ quartile class
$w$ = Class width

First, let's prepare the data by finding the class boundaries and cumulative frequencies. The given class intervals are discrete, so we convert them to continuous class boundaries by subtracting 0.5 from the lower limit and adding 0.5 to the upper limit.

Class Interval	Class Boundaries	Frequency (f)	Cumulative Frequency (cf)
20-21	19.5-21.5	4	4
22-23	21.5-23.5	12	4 + 12 = 16
24-25	23.5-25.5	24	16 + 24 = 40
26-27	25.5-27.5	12	40 + 12 = 52
28-29	27.5-29.5	4	52 + 4 = 56

From the table:
Total frequency $n = \sum f = 56$ .
Class width $w = 21.5 - 19.5 = 2$ .

Calculate the First Quartile ( $Q_1$ ).
The position of $Q_1$ is $\frac{1n}{4} = \frac{1 \times 56}{4} = 14^{th}$ item.
The $14^{th}$ item falls in the class with a cumulative frequency of 16, which is the 22-23 class (class boundaries 21.5-23.5).
For this class:
- $L = 21.5$
- $cfb = 4$ (cumulative frequency of the preceding class)
- $f = 12$ (frequency of the $Q_1$ class)
- $w = 2$
Substitute these values into the formula:
$Q_1 = 21.5 + \left(\frac{14 - 4}{12}\right)2$ $Q_1 = 21.5 + \left(\frac{10}{12}\right)2$ $Q_1 = 21.5 + \frac{20}{12}$ $Q_1 = 21.5 + 1.666...$ $Q_1 \approx 23.17$
Calculate the Third Quartile ( $Q_3$ ).
The position of $Q_3$ is $\frac{3n}{4} = \frac{3 \times 56}{4} = 3 \times 14 = 42^{nd}$ item.
The $42^{nd}$ item falls in the class with a cumulative frequency of 52, which is the 26-27 class (class boundaries 25.5-27.5).
For this class:
- $L = 25.5$
- $cfb = 40$ (cumulative frequency of the preceding class)
- $f = 12$ (frequency of the $Q_3$ class)
- $w = 2$
Substitute these values into the formula:
$Q_3 = 25.5 + \left(\frac{42 - 40}{12}\right)2$ $Q_3 = 25.5 + \left(\frac{2}{12}\right)2$ $Q_3 = 25.5 + \frac{4}{12}$ $Q_3 = 25.5 + 0.333...$ $Q_3 \approx 25.83$
Calculate the Interquartile Range (IQR).
The Interquartile Range is the difference between the third and first quartiles:
$IQR = Q_3 - Q_1$ $IQR = 25.83 - 23.17$ $IQR = 2.66$
Calculate the Quartile Deviation (QD).
The Quartile Deviation is half of the Interquartile Range:
$QD = \frac{IQR}{2}$ $QD = \frac{2.66}{2}$ $QD = 1.33$
The quartile deviation is $\text{1.33}$ .

Calculate the quartile deviation for the given grouped data.

Answer

Step-by-step solution

Calculate the First Quartile ( $Q_1$ ).

Calculate the Third Quartile ( $Q_3$ ).

Calculate the Interquartile Range (IQR).

Calculate the Quartile Deviation (QD).

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Calculate the quartile deviation for the given grouped data.

Answer

Step-by-step solution

Calculate the First Quartile (Q1Q_1Q1​).

Calculate the Third Quartile (Q3Q_3Q3​).

Calculate the Interquartile Range (IQR).

Calculate the Quartile Deviation (QD).

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Calculate the First Quartile ( $Q_1$ ).

Calculate the Third Quartile ( $Q_3$ ).