Sec 4 · G3 (Express)

Statistics

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1. Cumulative frequency

Marks	Frequency	Cumulative frequency
$0 < x \leq 10$	$2$	$2$
$10 < x \leq 20$	$6$	$8$
$20 < x \leq 30$	$14$	$22$
$30 < x \leq 40$	$22$	$44$
$40 < x \leq 50$	$11$	$55$
$50 < x \leq 60$	$5$	$60$

Cumulative frequency curve for 60 students, plotted at the upper end of each class. Reading across from 15, 30 and 45 gives the lower quartile about 25.7, the median about 33.7 and the upper quartile about 40.6.

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students in a test. Use the curve to estimate the number of students who scored more than $15$ marks.

Step 1 of 2

Go up from the mark on the horizontal axis to the curve, then across

The curve gives about

4

students with at most

15

marks.

Practise this →

2. Median, quartiles and percentiles

From a curve ( $60$ values)	Read across from	Here
lower quartile $Q_1$	$\dfrac{1}{4} \times 60$ , so $15$	$25.7$
median	$\dfrac{1}{2} \times 60$ , so $30$	$33.7$
upper quartile $Q_3$	$\dfrac{3}{4} \times 60$ , so $45$	$40.6$
interquartile range	$Q_3 - Q_1$	$14.9$
$p$ th percentile	$\dfrac{p}{100} \times 60$	80th: across from $48$

From a list: 39, 42, 44, 46, 48, 50, 52, 55, 57, 61

\displaystyle n = 10

median: between the 5th and 6th, $\dfrac{48 + 50}{2}$ , so $49$
lower half $39, 42, \mathbf{44}, 46, 48$ : $Q_1 = 44$
upper half $50, 52, \mathbf{55}, 57, 61$ : $Q_3 = 55$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. Use the curve to estimate the median mark.

Step 1 of 2

Which cumulative frequency do you go across from?

\mathstrut \dfrac{60}{2} = 30

Asked	On the curve
percentage scoring $x$ or fewer	up from $x$ , across: reading $\div N \times 100\%$
$t$ with $P(\text{more than } t) = q$	across from $(1 - q)N$ , then down

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. The probability that a student chosen at random scored more than $t$ marks is $0.2$ . Use the curve to estimate $t$ .

Step 1 of 3

Turn the probability into a number of students

More than

t

marks:

0.2 \times 60 = 12

students

Practise this →

3. Box-and-whisker plots

\displaystyle \text{min} \;\; Q_1 \;\; \text{median} \;\; Q_3 \;\; \text{max}

Measure	Formula	Here
range	max $-$ min	$36 - 8$ , so $28$
interquartile range	$Q_3 - Q_1$	$27 - 16$ , so $11$

📖 Worked example

The box-and-whisker plots show the times, in minutes, taken by two groups, $A$ and $B$ . Find the difference between their medians.

Step 1 of 2

Read each plot

A

\text{min} = 18

Q_1 = 24

\text{median} = 30

Q_3 = 38

\text{max} = 50

B

\text{min} = 22

Q_1 = 30

\text{median} = 34

Q_3 = 42

\text{max} = 48

📖 Worked example

The box-and-whisker plot shows the times, in minutes, that some students took to finish a task. $50$ of the students took less than $20$ minutes. Find the number of students who did the task.

Step 1 of 2

The lower whisker (minimum to Q1 = 20) holds a quarter of the data

A quarter of the students took less than

20

minutes.

Practise this →

4. Mean and standard deviation

	List	Frequency table
mean	$\bar{x} = \dfrac{\Sigma x}{n}$	$\bar{x} = \dfrac{\Sigma fx}{\Sigma f}$
standard deviation	$\sqrt{\dfrac{\Sigma x^{2}}{n} - \left(\dfrac{\Sigma x}{n}\right)^{2}}$	$\sqrt{\dfrac{\Sigma fx^{2}}{\Sigma f} - \left(\dfrac{\Sigma fx}{\Sigma f}\right)^{2}}$
or	$\sqrt{\dfrac{\Sigma (x - \bar{x})^{2}}{n}}$	$\sqrt{\dfrac{\Sigma f(x - \bar{x})^{2}}{\Sigma f}}$

30, 34, 35, 39, 42: Σx = 180, Σx² = 6566

\displaystyle \sqrt{\dfrac{\Sigma x^{2}}{n} - \left(\dfrac{\Sigma x}{n}\right)^{2}}

e.g.

\displaystyle \sqrt{\dfrac{6566}{5} - \left(\dfrac{180}{5}\right)^{2}}

\displaystyle = 4.15\text{ (3 s.f.)}

The same data, the other way: mean 36

\displaystyle \sqrt{\dfrac{\Sigma (x - \bar{x})^{2}}{n}}

e.g.

\displaystyle \sqrt{\dfrac{(-6)^{2} + (-2)^{2} + (-1)^{2} + 3^{2} + 6^{2}}{5}}

\displaystyle = 4.15\text{ (3 s.f.)}

📖 Worked example

The table shows the marks of $60$ students. Using the mid-values of the classes, estimate the mean mark, correct to 3 significant figures.

Marks, $x$	Frequency
$0 < x \le 10$	2
$10 < x \le 20$	6
$20 < x \le 30$	14
$30 < x \le 40$	22
$40 < x \le 50$	11
$50 < x \le 60$	5

Step 1 of 2

Use the mid-values x with the frequencies f

x = 5, 15, 25, 35, 45, 55

\Sigma f = 60

\Sigma fx = 1990

Every value	Mean	Standard deviation
$+\ c$ or $-\ c$	$+\ c$ or $-\ c$	unchanged
$\times\ k$	$\times\ k$	$\times\ k$

📖 Worked example

The mean and the standard deviation of the marks of a class in a test are $32.4$ marks and $6.8$ marks. Every student is then given $3$ extra marks for a question that was marked wrongly. Write down the new standard deviation.

Step 1 of 2

Moving every value by the same amount moves the mean, but the spread about it does not change

The standard deviation stays the same.

Practise this →

5. Comparing two data sets

Compare	Meaning
mean or median	larger: the values are higher on average (say what that means, e.g. better marks, or took longer)
standard deviation or IQR	smaller: more consistent (less spread out)

📖 Worked example

The numbers of books read by some students in two classes are shown. Class $A$ : $25, \ 30, \ 28, \ 22, \ 35$ . Class $B$ : $12, \ 15, \ 14, \ 13, \ 16$ . Which class's numbers of books read were more consistent?

Step 1 of 4

Which measure answers the question?

More consistent means less spread: the smaller standard deviation.

Practise this →

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Sec 4 · G3 (Express)

Statistics

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1. Cumulative frequency

Marks	Frequency	Cumulative frequency
$0 < x \leq 10$	$2$	$2$
$10 < x \leq 20$	$6$	$8$
$20 < x \leq 30$	$14$	$22$
$30 < x \leq 40$	$22$	$44$
$40 < x \leq 50$	$11$	$55$
$50 < x \leq 60$	$5$	$60$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students in a test. Use the curve to estimate the number of students who scored more than $15$ marks.

Step 1 of 2

Go up from the mark on the horizontal axis to the curve, then across

The curve gives about

4

students with at most

15

marks.

Practise this →

2. Median, quartiles and percentiles

From a curve ( $60$ values)	Read across from	Here
lower quartile $Q_1$	$\dfrac{1}{4} \times 60$ , so $15$	$25.7$
median	$\dfrac{1}{2} \times 60$ , so $30$	$33.7$
upper quartile $Q_3$	$\dfrac{3}{4} \times 60$ , so $45$	$40.6$
interquartile range	$Q_3 - Q_1$	$14.9$
$p$ th percentile	$\dfrac{p}{100} \times 60$	80th: across from $48$

From a list: 39, 42, 44, 46, 48, 50, 52, 55, 57, 61

\displaystyle n = 10

median: between the 5th and 6th, $\dfrac{48 + 50}{2}$ , so $49$
lower half $39, 42, \mathbf{44}, 46, 48$ : $Q_1 = 44$
upper half $50, 52, \mathbf{55}, 57, 61$ : $Q_3 = 55$

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. Use the curve to estimate the median mark.

Step 1 of 2

Which cumulative frequency do you go across from?

\mathstrut \dfrac{60}{2} = 30

Asked	On the curve
percentage scoring $x$ or fewer	up from $x$ , across: reading $\div N \times 100\%$
$t$ with $P(\text{more than } t) = q$	across from $(1 - q)N$ , then down

📖 Worked example

The cumulative frequency curve shows the marks of $60$ students. The probability that a student chosen at random scored more than $t$ marks is $0.2$ . Use the curve to estimate $t$ .

Step 1 of 3

Turn the probability into a number of students

More than

t

marks:

0.2 \times 60 = 12

students

Practise this →

3. Box-and-whisker plots

\displaystyle \text{min} \;\; Q_1 \;\; \text{median} \;\; Q_3 \;\; \text{max}

Measure	Formula	Here
range	max $-$ min	$36 - 8$ , so $28$
interquartile range	$Q_3 - Q_1$	$27 - 16$ , so $11$

📖 Worked example

The box-and-whisker plots show the times, in minutes, taken by two groups, $A$ and $B$ . Find the difference between their medians.

Step 1 of 2

Read each plot

A

\text{min} = 18

Q_1 = 24

\text{median} = 30

Q_3 = 38

\text{max} = 50

B

\text{min} = 22

Q_1 = 30

\text{median} = 34

Q_3 = 42

\text{max} = 48

📖 Worked example

The box-and-whisker plot shows the times, in minutes, that some students took to finish a task. $50$ of the students took less than $20$ minutes. Find the number of students who did the task.

Step 1 of 2

The lower whisker (minimum to Q1 = 20) holds a quarter of the data

A quarter of the students took less than

20

minutes.

Practise this →

4. Mean and standard deviation

	List	Frequency table
mean	$\bar{x} = \dfrac{\Sigma x}{n}$	$\bar{x} = \dfrac{\Sigma fx}{\Sigma f}$
standard deviation	$\sqrt{\dfrac{\Sigma x^{2}}{n} - \left(\dfrac{\Sigma x}{n}\right)^{2}}$	$\sqrt{\dfrac{\Sigma fx^{2}}{\Sigma f} - \left(\dfrac{\Sigma fx}{\Sigma f}\right)^{2}}$
or	$\sqrt{\dfrac{\Sigma (x - \bar{x})^{2}}{n}}$	$\sqrt{\dfrac{\Sigma f(x - \bar{x})^{2}}{\Sigma f}}$

30, 34, 35, 39, 42: Σx = 180, Σx² = 6566

\displaystyle \sqrt{\dfrac{\Sigma x^{2}}{n} - \left(\dfrac{\Sigma x}{n}\right)^{2}}

e.g.

\displaystyle \sqrt{\dfrac{6566}{5} - \left(\dfrac{180}{5}\right)^{2}}

\displaystyle = 4.15\text{ (3 s.f.)}

The same data, the other way: mean 36

\displaystyle \sqrt{\dfrac{\Sigma (x - \bar{x})^{2}}{n}}

e.g.

\displaystyle \sqrt{\dfrac{(-6)^{2} + (-2)^{2} + (-1)^{2} + 3^{2} + 6^{2}}{5}}

\displaystyle = 4.15\text{ (3 s.f.)}

📖 Worked example

The table shows the marks of $60$ students. Using the mid-values of the classes, estimate the mean mark, correct to 3 significant figures.

Marks, $x$	Frequency
$0 < x \le 10$	2
$10 < x \le 20$	6
$20 < x \le 30$	14
$30 < x \le 40$	22
$40 < x \le 50$	11
$50 < x \le 60$	5

Step 1 of 2

Use the mid-values x with the frequencies f

x = 5, 15, 25, 35, 45, 55

\Sigma f = 60

\Sigma fx = 1990

Every value	Mean	Standard deviation
$+\ c$ or $-\ c$	$+\ c$ or $-\ c$	unchanged
$\times\ k$	$\times\ k$	$\times\ k$

📖 Worked example

Step 1 of 2

Moving every value by the same amount moves the mean, but the spread about it does not change

The standard deviation stays the same.

Practise this →

5. Comparing two data sets

Compare	Meaning
mean or median	larger: the values are higher on average (say what that means, e.g. better marks, or took longer)
standard deviation or IQR	smaller: more consistent (less spread out)

📖 Worked example

Step 1 of 4

Which measure answers the question?

More consistent means less spread: the smaller standard deviation.

Practise this →

Try a question: Statistics

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

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