Question

Find the upper quartile for the following distribution is given by the size of

Size of Items	1	2	3	4	5	6	7
Frequency	2	4	5	8	7	3	2

A. ${\left( {\dfrac{{31 + 1}}{4}} \right)^{th}}$ term
B. ${\left[ {2\left( {\dfrac{{31 + 1}}{4}} \right)} \right]^{th}}$ term
C. ${\left[ {3\left( {\dfrac{{31 + 1}}{4}} \right)} \right]^{th}}$ term
D. ${\left[ {4\left( {\dfrac{{31 + 1}}{4}} \right)} \right]^{th}}$ term

Answer 1

Hint: First, calculate the cumulative frequency of the given data set by adding the given frequencies. Then calculate the number of total observations. In the end, substitute the value of the total observations in the formula of the upper quartile to get the required answer.

Formula Used:
The formula of an upper quartile of a data set: ${Q_3} = {\left[ {3\left( {\dfrac{{n + 1}}{4}} \right)} \right]^{th}}$ term, where n is the number of observations

Complete step by step solution:
The given distribution of values is,

Size of Items	1	2	3	4	5	6	7
Frequency	2	4	5	8	7	3	2

Let’s calculate the cumulative frequency of the above data.

Size of Items	1	2	3	4	5	6	7
Frequency	2	4	5	8	7	3	2
Cumulative Frequency	2	6	11	19	26	29	31

From the above data table, we get
The total number of observations: $31$
Now apply the formula of an upper quartile ${Q_3} = {\left[ {3\left( {\dfrac{{n + 1}}{4}} \right)} \right]^{th}}$ term.
Substitute $n = 31$ in the above formula.
We get,
${Q_3} = {\left[ {3\left( {\dfrac{{31 + 1}}{4}} \right)} \right]^{th}}$ term

Option ‘C’ is correct

Additional Information:
The quartiles of a data set are the numbers used to divide a set of data into four equal parts, or quarters. The upper quartile is calculated by determining the median of the upper half of a data set.
Steps to find the upper quartile:
Step 1: Arrange the numbers of the data set in ascending order.
Step 2: Calculate how many numbers are in the data set.
Step 3: Use the formula ${Q_3} = \dfrac{3}{4}{\left( {n + 1} \right)^{th}}term$ to calculate the upper quartile.

Note: Students often get confused about the formulas of the first, second, and third quartiles.
Following are the formula of the quartiles:
First quartile: ${Q_1} = \dfrac{1}{4}{\left( {n + 1} \right)^{th}}term$
Second quartile: ${Q_2} = \dfrac{1}{2}{\left( {n + 1} \right)^{th}}term$
Third quartile: ${Q_3} = \dfrac{3}{4}{\left( {n + 1} \right)^{th}}term$