Data & Statistics — Level 2 — Functional Skills Maths

Grouped frequency tables and estimated mean

When data is grouped into classes (e.g. 0–9, 10–19…), you cannot find the exact mean. Instead, use the midpoint of each class as a representative value.

Estimated mean = Σ(midpoint × frequency) ÷ Σ(frequency)
Midpoint = (lower bound + upper bound) ÷ 2

💡 Modal class: The group with the highest frequency is called the modal class — not a single value like the mode.

Scatter graphs and correlation

A scatter graph plots two variables against each other to look for a relationship. The strength and direction of the relationship is called correlation.

Positive

Both increase together

Negative

One increases, other decreases

None

No pattern / scattered

Line of best fit

Drawn through the middle

💡 Line of best fit: Should pass through approximately the mean point (x̄, ȳ) with roughly equal numbers of points on each side.

Two-way tables

A two-way table organises data by two categories simultaneously, showing the frequency for each combination. Row totals and column totals must all add up to the grand total.

💡 Missing values: Use totals to find missing cells. If row total = 50 and one cell = 32, the other cell = 50 − 32 = 18.

Combined probability

When two events happen independently, you can find the probability of both occurring (AND) or either occurring (OR).

P(A AND B) = P(A) × P(B) ← independent events
P(A OR B) = P(A) + P(B) − P(A AND B)
For mutually exclusive events: P(A OR B) = P(A) + P(B)

AND

Multiply probabilities

Add (then subtract overlap)

Independent

One event doesn't affect other

💡 Tree diagrams are useful for combined probability — they help you track AND (multiply along branches) and OR (add the final outcomes you want).

1Estimated mean from grouped data

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Estimate the mean age from this grouped frequency table:

Age group	Midpoint	Frequency	Mid × Freq
20–29	24.5	8	196
30–39	34.5	12	414
40–49	44.5	10	445
50–59	54.5	5	272.5
Total		35	1327.5

1Estimated mean: 1327.5 ÷ 35 = 37.9

✅ Estimated mean age ≈ 37.9 years

2Modal class and median class

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Using the age table above, identify (a) the modal class and (b) the class containing the median.

1(a) Highest frequency = 12, which is in the 30–39 group. Modal class = 30–39.
2(b) Total = 35 people. Median is the 18th value. Cumulative: first 8 (20–29), first 20 (30–39). The 18th value falls in 30–39.

✅ Modal class = 30–39 | Median class = 30–39

3Reading a two-way table

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Complete the missing value and answer the question:

	Tea	Coffee	Total
Male	18	?	35
Female	22	13	35
Total	40	30	70

1Missing value: Male coffee = total males − male tea = 35 − 18 = 17
2Check: total coffee = 17 + 13 = 30 ✓

✅ Missing value = 17

4Combined probability (AND)

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A coin is flipped and a dice is rolled. What is the probability of getting heads AND a 6?

1P(heads) = 1/2
2P(6) = 1/6
3Independent events: P(heads AND 6) = 1/2 × 1/6 = 1/12

✅ P(heads AND 6) = 1/12 ≈ 0.083

5Combined probability — tree diagram

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A bag has 3 red and 2 blue counters. One is drawn and replaced. Then another is drawn. Find P(both same colour).

1P(red) = 3/5, P(blue) = 2/5 (with replacement — same each time)
2P(RR) = 3/5 × 3/5 = 9/25
3P(BB) = 2/5 × 2/5 = 4/25
4P(same) = 9/25 + 4/25 = 13/25 = 0.52

✅ P(both same colour) = 13/25 = 0.52

6Describing correlation

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A scatter graph plots temperature (x-axis) against ice cream sales (y-axis). The points cluster along a rising line from bottom-left to top-right. Describe the correlation and use the line of best fit to estimate sales at 28°C if the line passes through (20, 150) and (30, 250).

1Rising from left to right = positive correlation. As temperature increases, ice cream sales increase.
2Gradient: (250−150)÷(30−20) = 100÷10 = 10 sales per °C
3At 28°C: 150 + (10 × 8) = 150 + 80 = 230 sales

✅ Strong positive correlation | Estimated sales at 28°C = 230

Question 1

Grouped data: 0–9 (freq 4, mid 4.5), 10–19 (freq 6, mid 14.5), 20–29 (freq 10, mid 24.5).

What is the estimated mean?

Question 2

Two-way table: 100 students. Boys who like sport: 28. Girls who like sport: 32. Total who like sport: 60.

How many girls do NOT like sport?

Question 2 (corrected)

A two-way table: 80 people total. 35 are male. 48 prefer cinema. 20 males prefer cinema. How many females prefer cinema?

Question 3

P(A) = 0.4, P(B) = 0.3. A and B are independent events. What is P(A and B)?

Question 4

A scatter graph shows temperature vs heating bills. Points cluster from top-left to bottom-right. What type of correlation is this?

Question 5

A spinner has P(red) = 0.3, P(blue) = 0.5, P(green) = 0.2. It is spun twice independently. What is P(red then blue)?

Question 6

Grouped data: 1–5 (f=3, mid=3), 6–10 (f=7, mid=8), 11–15 (f=5, mid=13). What is the modal class?

Question 7

A bag has 4 red and 6 blue balls. One ball is taken, replaced, then another is taken. What is P(both red)?

Question 8

P(pass exam) = 0.75. Two students take it independently. What is P(both pass)?

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