Sec 2 · G3 (Express)

Statistics

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1. Dot diagrams, stem-and-leaf diagrams and histograms

A dot diagram of books read (3, 4, 4, 5, 5, 5, 6, 6, 7, 9), and a histogram of times in classes of 10 minutes with frequencies 3, 7, 12, 6 and 2, bars touching.

Stem	Leaf
$2$	$3\ 5\ 8$
$3$	$0\ 4\ 4\ 7$
$4$	$1\ 6$
Key: $3 \mid 4$ means $34$

📖 Worked example

The dot diagram shows the number of hours some pupils slept last night. Find the median.

Step 1 of 2

Count the dots: each dot is one value

Number of values

= 14

2A leaves	Stem	2B leaves
$8\ 3$	$4$	$1\ 5\ 9$
$7\ 6\ 2$	$5$	$0\ 4$
$9\ 4\ 1$	$6$	$2\ 2\ 8$
Key: $3 \mid 4 \mid 1$ means $43$ for 2A and $41$ for 2B

📖 Worked example

The back-to-back stem-and-leaf diagram shows the heights, in cm, of seedlings grown with fertiliser X and with fertiliser Y. By comparing the ranges, state which group's results are more consistent.

Step 1 of 2

Find each group's range: largest − smallest

Range of X

= 58 - 28

= 30

Range of Y

= 62 - 29

= 33

📖 Worked example

The stem-and-leaf diagram shows the ages of the people at a community centre class. One leaf is shown as $x$ . The median is $41$ years. Find the value of $x$ .

Step 1 of 2

Count the leaves (x is one of them): where is the median?

There are

12

values.The median is the mean of the

6

th and

7

th values:

42

and

40 + x

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2. Mean, median and mode

Average	Meaning	Books read: $3, 4, 4, 5, 5, 5, 6, 6, 7, 9$
mean	$\dfrac{\text{sum}}{\text{number of values}}$	$\dfrac{54}{10}$ , so $5.4$
median	the middle value, in order	the 5th and 6th are both $5$ : $5$
mode	the most common value	$5$

Mean from a frequency table: Σfx ÷ Σf

\displaystyle \dfrac{1(2) + 2(5) + 3(3)}{2 + 5 + 3}

e.g.

\displaystyle \dfrac{1(2) + 2(5) + 3(3)}{2 + 5 + 3}

\displaystyle = 2.1

📖 Worked example

The numbers of books read by 6 pupils last month are $3,\ 0,\ 8,\ 4,\ x,\ 1$ . The mean is $3$ . Find $x$ .

Step 1 of 3

Total of all the values = mean × number of values

Total

= 3 \times 6

= 18

📖 Worked example

The frequency table shows the number of co-curricular activities each pupil takes part in. Find the mean.

Number of CCAs	1	2	3	4	5	6
Frequency	4	9	4	3	3	5

Step 1 of 3

Total frequency: add the frequencies

Total frequency

= 4 + 9 + 4 + 3 + 3 + 5

= 28

Every value	Mean, median, mode	Range
$+\ 5$	$+\ 5$	unchanged
$-\ 2$	$-\ 2$	unchanged
$\times\ 3$	$\times\ 3$	$\times\ 3$

Which average?	When
median	one value is far from the rest (it pulls the mean towards it)
mode	the most common item or size is wanted, or the data are not numbers
mean	every value should count and none is extreme

📖 Worked example

The mean, median, mode and range of the marks scored by 12 pupils in a 10-mark quiz are $5.9$ marks, $6.5$ marks, $8$ marks and $7$ marks. Every mark is then multiplied by $2$ to give a mark out of $20$ . Find the new range.

Step 1 of 2

Multiplying every value by 2 multiplies every average AND the range by 2

Every value is multiplied by

2

, so the range is multiplied by

2

📖 Worked example

The list shows the monthly salaries, in dollars, of the 9 people working in a small company: $2450,\ 2950,\ 2300,\ 2800,\ 9800,\ 2700,\ 3200,\ 2600,\ 3100$ . Which average, the mean, the median or the mode, is the most suitable to use to describe a typical value?

Step 1 of 3

Work out the mean and the median

Mean

= \dfrac{31900}{9}

= 3544

(3 s.f.)Median

= 2800

📖 Worked example

Five whole numbers have a mode of $5$ , a median of $8$ , a mean of $8$ and a range of $7$ . Find the five numbers, from the smallest to the largest.

Step 1 of 4

The median is the middle (3rd) number

The 3rd number is the median,

8

📖 Worked example

The frequency table shows the number of co-curricular activities each pupil takes part in. The median is $3$ . Find the largest possible value of $k$ .

Number of CCAs	1	2	3	4	5	6
Frequency	6	2	2	2	5	k

Step 1 of 2

k values lie above 3: count the values on each side

Values above

3

k + 7

Values equal to

3 or less

10

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3. Grouped data

Time (min)	Mid-value $x$	Frequency $f$	$fx$
$0 < t \leq 10$	$5$	$3$	$15$
$10 < t \leq 20$	$15$	$7$	$105$
$20 < t \leq 30$	$25$	$12$	$300$
$30 < t \leq 40$	$35$	$6$	$210$
$40 < t \leq 50$	$45$	$2$	$90$
Total		$30$	$720$

Estimated mean: use the mid-values

\displaystyle \dfrac{\Sigma fx}{\Sigma f}

e.g.

\displaystyle \dfrac{720}{30}

\displaystyle = 24

📖 Worked example

The table shows the distances some members of a cycling club rode on Sunday. Using the mid-value of each class, calculate an estimate of the mean, correct to 3 significant figures.

Distance (d km)	Frequency
$0 < d \le 20$	12
$20 < d \le 40$	9
$40 < d \le 60$	6
$60 < d \le 80$	4

Step 1 of 4

Mid-value of each class: halfway between its ends

Mid-values:

10,\ 30,\ 50,\ 70

📖 Worked example

The table shows the masses of some parcels at a post office. Find the class that contains the median.

Mass (m kg)	Frequency
$10 < m \le 20$	9
$20 < m \le 30$	7
$30 < m \le 40$	8
$40 < m \le 50$	4
$50 < m \le 60$	2
$60 < m \le 70$	11

Step 1 of 3

Total frequency: where is the middle?

Total frequency

= 9 + 7 + 8 + 4 + 2 + 11

= 41

The median is the

\dfrac{41 + 1}{2} = 21

th value.

Practise this →

4. Histograms

To find	On the histogram
the frequency of a class	the height of its bar
the total frequency	add the heights of all the bars
the modal class	the class of the tallest bar
how many have $t > 30$	add the bars to the right of $30$

📖 Worked example

The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Find the number of members with $d > 30$ .

Step 1 of 2

Values more than 30: the classes from the boundary 30 upwards

d > 30

: the classes

30 < d \le 40

60 < d \le 70

(a value of exactly

30

is in the class below).

📖 Worked example

The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Using the mid-value of each class, calculate an estimate of the mean, correct to 3 significant figures.

Step 1 of 5

Read the height of each bar on the frequency axis

Frequencies:

12,\ 15,\ 5,\ 8,\ 5

Practise this →

5. Choosing a statistical diagram

Diagram	Best for	Limitation
Dot diagram	a small set of data with a few different values	too crowded for a large set of data
Stem-and-leaf diagram	keeping every value while showing the shape; the median is easy to find	needs a key; awkward for a very large set

Diagram	Best for	Limitation
Histogram	a large set of continuous data grouped in equal classes (bars touch)	the original values are lost

📖 Worked example

The marks out of 100 of 25 pupils in a test (from 31 to 98) are to be shown so that every mark is kept and the median is easy to find. Which statistical diagram is the most suitable?

Step 1 of 2

What must the diagram do?

The diagram has to keep every value while showing the shape of the data.

Practise this →

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Sec 2 · G3 (Express)

Statistics

Practise statistics with step-by-step help. Free account.

Start free Worksheet (PDF)

1. Dot diagrams, stem-and-leaf diagrams and histograms

Stem	Leaf
$2$	$3\ 5\ 8$
$3$	$0\ 4\ 4\ 7$
$4$	$1\ 6$
Key: $3 \mid 4$ means $34$

📖 Worked example

The dot diagram shows the number of hours some pupils slept last night. Find the median.

Step 1 of 2

Count the dots: each dot is one value

Number of values

= 14

2A leaves	Stem	2B leaves
$8\ 3$	$4$	$1\ 5\ 9$
$7\ 6\ 2$	$5$	$0\ 4$
$9\ 4\ 1$	$6$	$2\ 2\ 8$
Key: $3 \mid 4 \mid 1$ means $43$ for 2A and $41$ for 2B

📖 Worked example

The back-to-back stem-and-leaf diagram shows the heights, in cm, of seedlings grown with fertiliser X and with fertiliser Y. By comparing the ranges, state which group's results are more consistent.

Step 1 of 2

Find each group's range: largest − smallest

Range of X

= 58 - 28

= 30

Range of Y

= 62 - 29

= 33

📖 Worked example

The stem-and-leaf diagram shows the ages of the people at a community centre class. One leaf is shown as $x$ . The median is $41$ years. Find the value of $x$ .

Step 1 of 2

Count the leaves (x is one of them): where is the median?

There are

12

values.The median is the mean of the

6

th and

7

th values:

42

and

40 + x

Practise this →

2. Mean, median and mode

Average	Meaning	Books read: $3, 4, 4, 5, 5, 5, 6, 6, 7, 9$
mean	$\dfrac{\text{sum}}{\text{number of values}}$	$\dfrac{54}{10}$ , so $5.4$
median	the middle value, in order	the 5th and 6th are both $5$ : $5$
mode	the most common value	$5$

Mean from a frequency table: Σfx ÷ Σf

\displaystyle \dfrac{1(2) + 2(5) + 3(3)}{2 + 5 + 3}

e.g.

\displaystyle \dfrac{1(2) + 2(5) + 3(3)}{2 + 5 + 3}

\displaystyle = 2.1

📖 Worked example

The numbers of books read by 6 pupils last month are $3,\ 0,\ 8,\ 4,\ x,\ 1$ . The mean is $3$ . Find $x$ .

Step 1 of 3

Total of all the values = mean × number of values

Total

= 3 \times 6

= 18

📖 Worked example

The frequency table shows the number of co-curricular activities each pupil takes part in. Find the mean.

Number of CCAs	1	2	3	4	5	6
Frequency	4	9	4	3	3	5

Step 1 of 3

Total frequency: add the frequencies

Total frequency

= 4 + 9 + 4 + 3 + 3 + 5

= 28

Every value	Mean, median, mode	Range
$+\ 5$	$+\ 5$	unchanged
$-\ 2$	$-\ 2$	unchanged
$\times\ 3$	$\times\ 3$	$\times\ 3$

Which average?	When
median	one value is far from the rest (it pulls the mean towards it)
mode	the most common item or size is wanted, or the data are not numbers
mean	every value should count and none is extreme

📖 Worked example

Step 1 of 2

Multiplying every value by 2 multiplies every average AND the range by 2

Every value is multiplied by

2

, so the range is multiplied by

2

📖 Worked example

Step 1 of 3

Work out the mean and the median

Mean

= \dfrac{31900}{9}

= 3544

(3 s.f.)Median

= 2800

📖 Worked example

Five whole numbers have a mode of $5$ , a median of $8$ , a mean of $8$ and a range of $7$ . Find the five numbers, from the smallest to the largest.

Step 1 of 4

The median is the middle (3rd) number

The 3rd number is the median,

8

📖 Worked example

The frequency table shows the number of co-curricular activities each pupil takes part in. The median is $3$ . Find the largest possible value of $k$ .

Number of CCAs	1	2	3	4	5	6
Frequency	6	2	2	2	5	k

Step 1 of 2

k values lie above 3: count the values on each side

Values above

3

k + 7

Values equal to

3 or less

10

Practise this →

3. Grouped data

Time (min)	Mid-value $x$	Frequency $f$	$fx$
$0 < t \leq 10$	$5$	$3$	$15$
$10 < t \leq 20$	$15$	$7$	$105$
$20 < t \leq 30$	$25$	$12$	$300$
$30 < t \leq 40$	$35$	$6$	$210$
$40 < t \leq 50$	$45$	$2$	$90$
Total		$30$	$720$

Estimated mean: use the mid-values

\displaystyle \dfrac{\Sigma fx}{\Sigma f}

e.g.

\displaystyle \dfrac{720}{30}

\displaystyle = 24

📖 Worked example

The table shows the distances some members of a cycling club rode on Sunday. Using the mid-value of each class, calculate an estimate of the mean, correct to 3 significant figures.

Distance (d km)	Frequency
$0 < d \le 20$	12
$20 < d \le 40$	9
$40 < d \le 60$	6
$60 < d \le 80$	4

Step 1 of 4

Mid-value of each class: halfway between its ends

Mid-values:

10,\ 30,\ 50,\ 70

📖 Worked example

The table shows the masses of some parcels at a post office. Find the class that contains the median.

Mass (m kg)	Frequency
$10 < m \le 20$	9
$20 < m \le 30$	7
$30 < m \le 40$	8
$40 < m \le 50$	4
$50 < m \le 60$	2
$60 < m \le 70$	11

Step 1 of 3

Total frequency: where is the middle?

Total frequency

= 9 + 7 + 8 + 4 + 2 + 11

= 41

The median is the

\dfrac{41 + 1}{2} = 21

th value.

Practise this →

4. Histograms

To find	On the histogram
the frequency of a class	the height of its bar
the total frequency	add the heights of all the bars
the modal class	the class of the tallest bar
how many have $t > 30$	add the bars to the right of $30$

📖 Worked example

The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Find the number of members with $d > 30$ .

Step 1 of 2

Values more than 30: the classes from the boundary 30 upwards

d > 30

: the classes

30 < d \le 40

60 < d \le 70

(a value of exactly

30

is in the class below).

📖 Worked example

The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Using the mid-value of each class, calculate an estimate of the mean, correct to 3 significant figures.

Step 1 of 5

Read the height of each bar on the frequency axis

Frequencies:

12,\ 15,\ 5,\ 8,\ 5

Practise this →

5. Choosing a statistical diagram

Diagram	Best for	Limitation
Dot diagram	a small set of data with a few different values	too crowded for a large set of data
Stem-and-leaf diagram	keeping every value while showing the shape; the median is easy to find	needs a key; awkward for a very large set

Diagram	Best for	Limitation
Histogram	a large set of continuous data grouped in equal classes (bars touch)	the original values are lost

📖 Worked example

The marks out of 100 of 25 pupils in a test (from 31 to 98) are to be shown so that every mark is kept and the median is easy to find. Which statistical diagram is the most suitable?

Step 1 of 2

What must the diagram do?

The diagram has to keep every value while showing the shape of the data.

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.

Practise statistics with step-by-step help

Endless questions from basic to exam level. Get one wrong and see where it went wrong. Free account.

Start free