The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below. Mark Totals 3 6 9 9 4 0 1 2 3 4 (6) 5 6 6 6 8 5 0 2 3 3 4 6 5 6 7 9 7 0 0 1 3 4 4 4 Key (3|6 means 36) (a) Write down the modal mark of these students. (b) Find the values of the lower quartile, the median and the upper quartile. For these students $\sum x = 2497$ and $\sum x^2 = 143 \ 369$ (c) Find the mean and the standard deviation of the marks of these students. (d) Describe the skewness of the marks of these students, giving a reason for your answer. The mean and standard deviation of the marks of all the students who sat the examination were 55 and 10 respectively. The examiners decided that the total mark of each student should be scaled by subtracting 5 and then reducing the mark by a further 10 %. (e) Find the mean and standard deviation of the scaled marks of all these students.

A-Level Edexcel Maths Statistics Data Presentation: The marks, x, of 45 students ran

The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 1

Question 4

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The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below. Mark Totals 3 6 9... show full transcript

Worked Solution & Example Answer:The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 1

Step 1

Write down the modal mark of these students.

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Answer

To find the modal mark, we need to identify the mark that appears most frequently. From the stem and leaf diagram, we see:

The mark 6 appears 6 times (in the totals under stem 6), and thus it is the most frequent mark.
Therefore, the modal mark is 6.

Step 2

Find the values of the lower quartile, the median and the upper quartile.

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Answer

To find the quartiles, we will first list the marks in ascending order:

36, 36, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 46, 46, 56, 56, 56, 58, 58, 66, 66, 66, 66, 67, 67, 69, 70, 70, 71, 73, 74, 75, 78, 79, 80, 81, 81, 81, 81, 81, 82, 82, 83, 84.
The total number of marks is 45.
Lower Quartile (Q1) is the 11th value: 46.
Median (Q2) is the 23rd value: 55.
Upper Quartile (Q3) is the 34th value: 66.

Step 3

Find the mean and the standard deviation of the marks of these students.

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Answer

The mean ( $\mu$ ) can be calculated using the formula:

$\mu = \frac{\sum x}{n} = \frac{2497}{45} \approx 55.49$

To find the standard deviation (sd), we use:

$sd = \sqrt{\frac{\sum x^2}{n} - (\mu)^2} = \sqrt{\frac{143369}{45} - (55.49)^2} \approx 10.45$

Step 4

Describe the skewness of the marks of these students, giving a reason for your answer.

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Answer

The relationship between the mean, median, and mode can be used to assess skewness:

Since $\text{mean} < \text{median} < \text{mode}$ , this indicates that the distribution is negatively skewed.
We conclude that there is a negative skewness in the data since the mean is less than the median.

Step 5

Find the mean and standard deviation of the scaled marks of all these students.

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Answer

First, we scale the marks. Scaling involves subtracting 5 from each mark and then further reducing this by 10%. The new mean can be calculated as:

$\text{New Mean} = (55 - 5) \times 0.9 = 45 \text{ (after scaling)}$

For the standard deviation:

$\text{New SD} = 10 \times 0.9 = 9$

The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 1

Worked Solution & Example Answer:The marks, x, of 45 students randomly selected from those students who sat a mathematics examination are shown in the stem and leaf diagram below - Edexcel - A-Level Maths Statistics - Question 4 - 2012 - Paper 1

Write down the modal mark of these students.

Find the values of the lower quartile, the median and the upper quartile.

Find the mean and the standard deviation of the marks of these students.

Describe the skewness of the marks of these students, giving a reason for your answer.

Find the mean and standard deviation of the scaled marks of all these students.

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