Question

find the median for the data items in the given frequency distribution. the median is
(type an integer or a decimal.)

Explanation:

Step1: Calculate the total number of data items

We use the formula \(n=\sum f\).

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Since \(n = 40\) (an even number), the median is the average of the \(\frac{n}{2}=20^{th}\) and \((\frac{n}{2}+ 1)=21^{st}\) ordered data items.

Step2: Find the cumulative frequencies

Score \(x\)	Frequency \(f\)	Cumulative Frequency \(cf\)
2	3	3 + 3=6
3	1	6+1 = 7
4	5	7+5 = 12
5	7	12+7 = 19
6	5	19+5 = 24
7	7	24+7 = 31
8	3	31+3 = 34
9	5	34+5 = 39
10	1	39+1 = 40

Step3: Locate the median - related data items

The \(20^{th}\) and \(21^{st}\) data items fall within the cumulative - frequency range where \(cf\geq20\). Looking at the cumulative - frequency column, when \(x = 6\), \(cf = 24\) (and before that, for \(x = 5\), \(cf=19\)). So both the \(20^{th}\) and \(21^{st}\) data items have a score of \(x = 6\).

Step4: Calculate the median

Since the \(20^{th}\) and \(21^{st}\) data items are both \(6\), the median \(M=\frac{6 + 6}{2}=6\)

Answer:

\(6\)