The table below gives the frequency distribution of the marks obtained by some students in a scholarship examination. Scores (x) 15 25 35 45 55 65 75 Freq (...
Question 1 Report
The table below gives the frequency distribution of the marks obtained by some students in a scholarship examination.
| Scores (x) | 15 | 25 | 35 | 45 | 55 | 65 | 75 |
| Freq (f) | 1 | 4 | 12 | 24 | 18 | 8 | 3 |
(a) Calculate, correct to 3 significant figures, the mean mark.
(b) Find the : (i) mode ; (ii) range of the distribution.
Answer Details
This is a discrete frequency distribution, so each score \(x\) already has its frequency \(f\). Build an \(fx\) column, add the columns, then apply the mean formula.
| Scores \(x\) | 15 | 25 | 35 | 45 | 55 | 65 | 75 | Total |
|---|---|---|---|---|---|---|---|---|
| Freq \(f\) | 1 | 4 | 12 | 24 | 18 | 8 | 3 | 70 |
| \(fx\) | 15 | 100 | 420 | 1080 | 990 | 520 | 225 | 3350 |
(a) Mean mark.
\[\bar{x} = \frac{\sum fx}{\sum f} = \frac{3350}{70} = 47.857\ldots\]Correct to 3 significant figures, the mean mark is \(\mathbf{47.9}\).
(b)(i) Mode. The mode is the score with the highest frequency. The largest frequency is \(24\), which belongs to the score \(45\), so the mode is \(\mathbf{45}\).
(b)(ii) Range.
\[\text{Range} = \text{highest score} - \text{lowest score} = 75 - 15 = \mathbf{60}.\]Examination note: multiply every score by its own frequency before adding, and divide by \(\sum f = 70\), not by the number of different scores (7). Dividing by 7 is the usual slip that gives a wrong mean near \(46\text{-}48\) instead of the correct \(47.9\).
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