Frequency tables questions with answers

Questions

1. Question 1The mean from a frequency table

   The frequency table shows the scores of pupils in a 10-mark spelling test. Find the mean, correct to 3 significant figures.

   Score	4	5	6	7	8	9
   Frequency	4	8	6	5	4	4

   Show the worked solution

   1. Step 1: Total frequency: add the frequenciesTotal frequency $= 4 + 8 + 6 + 5 + 4 + 4$$= 31$
   2. Step 2: Total of all the values: each value × its frequency, addedTotal of the values $= 4 \times 4 + 5 \times 8 + 6 \times 6 + 7 \times 5 + 8 \times 4 + 9 \times 4$$= 195$
   3. Step 3: Mean = total of the values ÷ total frequencyMean $= \dfrac{195}{31}$$= 6.2903\ldots$$= 6.29$ (3 s.f.)
2. Question 2The median from a frequency table

   The frequency table shows the scores of pupils in a 10-mark spelling test. Find the median.

   Score	4	5	6	7	8
   Frequency	9	5	3	2	2

   Show the worked solution

   1. Step 1: Total frequency: add the frequenciesTotal frequency $= 9 + 5 + 3 + 2 + 2$$= 21$
   2. Step 2: Running totals of the frequencies, from the smallest valueRunning totals: $9,\ 14,\ 17,\ 19,\ 21$
   3. Step 3: Find the position of the median, then the medianThe median is the $\dfrac{21 + 1}{2} = 11$th value.Median $= 5$
3. Question 3The mode from a frequency table

   The frequency table shows the number of people in each household in an HDB block. Find the mode.

   Number of people in the household	2	3	4	5	6	7
   Frequency	8	7	7	6	6	7

   Show the worked solution

   1. Step 1: Find the highest frequencyThe highest frequency is $8$.
   2. Step 2: The mode is the value with the highest frequency (not the frequency itself)Mode $= 2$
4. Question 4A missing frequency from the mean

   The frequency table shows the number of people in each household in an HDB block. The mean is $3.8$. Find $k$.

   Number of people in the household	1	2	3	4	5
   Frequency	1	2	1	6	k

   Show the worked solution

   1. Step 1: Total frequency and total of the values, with k for the unknown frequencyTotal frequency $= 10 + k$Total of the values $= 1 \times 1 + 2 \times 2 + 3 \times 1 + 4 \times 6 + 5k$$= 32 + 5k$
   2. Step 2: Mean = total of the values ÷ total frequency: write an equation in k$\dfrac{32 + 5k}{10 + k} = 3.8$$32 + 5k = 3.8(10 + k)$$32 + 5k = 38 + 3.8k$
   3. Step 3: Collect the k terms and solve$1.2k = 6$$k = 5$
5. Question 5The largest or smallest k for a given mode or median

   The frequency table shows the number of people in each household in an HDB block. The median is $6$. Find the smallest possible value of $k$.

   Number of people in the household	1	2	3	4	5	6
   Frequency	3	4	4	2	9	k

   Show the worked solution

   1. Step 1: Count the values below 6 and above 6Values below $6$: $22$Values above $6$: $0$
   2. Step 2: The median is the middle value: fewer than half of the values may lie on either side of it. With k values at 6, the bigger side must have fewer values than k and the other side together$22 < k + 0$$k > 22$Smallest $k = 23$