Question

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If the mean mark is 5.5
(i) find the value of y,
(ii) calculate the median and mode of the distribution.

Answer 1

Answer:

y = 9.

Median: 5.

Mode: 5.

Step-by-step explanation:

(i)

Express the mean of this distribution about $y$.

Multiple each mark by its frequency. Find the sum of these products. Divide by the sum of frequency.

Sum of products:

$2\times 9+8\times 6 + 5\;y + 3\times 6 + 4\times 8 + 4\;(y-2) + 6\times 7= 9\;y +150$.

Sum of frequencies:

$2 + 8 + y + 6 + 4 + (y-2) + 6 = 2\;y + 24$.

Mean:

$\displaystyle \frac{9\;y + 150}{2\;y +24} = 5.5$.

$11\;y + 132 = 9\;y + 150$.

$y = 9$.

(ii)

Rank the marks in an increasing order:

$\begin{array}{c|c}\text{Marks}&\text{Frequency}\\ 3&6 \\ 4&7\\ 5&9\\6&8\\7&6\\8&4\\9&2\end{array}$.

There are 42 terms. 42 / 2 = 21. What's the 21st mark? $6 + 7 < 21 < 6 + 7 + 9$. The 21st mark is a 5. That's going to be the median.

Median: 5.

The mode is the item with the greatest frequency.

Mode: 5, with a frequency of 9.