Frequency tables

Probability and statistics

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A group chooses colours: red $6$, blue $4$ and green $5$. How many people are there, and what is the mode?

2 marks
Shoe sizes $4, 5, 6, 7$ have frequencies $2, 5, 4, 1$. Find the mode and the number of people.

2 marks
The values $0, 1, 2, 3$ have frequencies $2, 3, 4, 1$. Find the mean.

3 marks
Goals in matches: $0, 1, 2, 3$ with frequencies $4, 6, 3, 2$. Find the mean number of goals and the median.

3 marks
Ages at a party: $6, 7, 8, 9$ with frequencies $3, 5, 8, 4$. Find the mode, the number of children and the mean age.

4 marks
The values $1, 2, 5$ have frequencies $4, 6, x$. The mean is $3$. Find $x$.

4 marks
A grouped table uses midpoints. The classes $0 \le t < 10$, $10 \le t < 20$, $20 \le t < 30$ and $30 \le t < 40$ have frequencies $4, 7, 6, 3$. Estimate the mean.

5 marks
Marks $5, 6, 7, 8, 9$ have frequencies $2, 5, 8, 4, 1$. Find the mode, median and mean. The pass mark is $8$. How many pass, and what percentage is that?

6 marks

Mark scheme

Answer. $15$ people, mode red
- $6 + 4 + 5 = 15$
- Red has the highest frequency.
Answer. Mode $5$, and $12$ people
- Size $5$ has the highest frequency.
- $2 + 5 + 4 + 1 = 12$
Answer. $1.4$
- Total $= 0 + 3 + 8 + 3 = 14$.
- Mean $= \dfrac{14}{10} = 1.4$.
Answer. Mean $1.2$, median $1$
- Total goals $= 6 + 6 + 6 = 18$, and there are $15$ matches.
- Mean $= \dfrac{18}{15} = 1.2$.
- The $8$th value is in the group of $1$s, so the median is $1$.
Answer. Mode $8$, $20$ children, mean $7.65$
- Age $8$ is the most common.
- Total $= 20$.
- Sum of ages $= 18 + 35 + 64 + 36 = 153$, so the mean is $7.65$.
Answer. $7$
- Sum $= 16 + 5x$ and the number of values is $10 + x$.
- $\dfrac{16 + 5x}{10 + x} = 3$ gives $2x = 14$.
- $x = 7$
Answer. $19$
- Midpoints $5, 15, 25, 35$.
- Total $= 20 + 105 + 150 + 105 = 380$.
- Estimated mean $= \dfrac{380}{20} = 19$.
Answer. Mode $7$, median $7$, mean $6.85$. Five people pass, which is $25\%$.
- The mode is $7$.
- Of $20$ values, the $10$th and $11$th are both $7$, so the median is $7$.
- The total of the marks is $137$, so the mean is $6.85$.
- Five of the $20$ marks are at least $8$, which is $25\%$.