Histograms questions with answers

Questions

1. Question 1Read a frequency off the histogram

   The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Write down the number of members in the class $40 < d \le 50$.

   Show the worked solution

   1. Step 1: Find the bar for the class on the horizontal axisThe class $40 < d \le 50$ is the bar from $40$ to $50$.
   2. Step 2: Read across from the top of the bar to the frequency axisFrequency $= 14$There are 14 members in this class.
2. Question 2Total frequency

   The histogram shows the time, $t$ minutes, that some pupils spent on homework last night. Find the total number of pupils.

   Show the worked solution

   1. Step 1: Read the height of each bar on the frequency axisFrequencies: $6,\ 12,\ 3,\ 15,\ 13,\ 5$
   2. Step 2: Total frequency: add the heights of all the barsTotal $= 6 + 12 + 3 + 15 + 13 + 5$$= 54$
3. Question 3The modal class

   The histogram shows the heights, $h$ cm, of some seedlings after two weeks. Write down the modal class.

   Show the worked solution

   1. Step 1: Find the tallest barThe tallest bar has frequency $11$.
   2. Step 2: The modal class is the class of the tallest barThe modal class is $4 < h \le 8$.
4. Question 4How many in a range

   The histogram shows the heights, $h$ cm, of some seedlings after two weeks. Find the number of seedlings with $12 < h \le 20$.

   Show the worked solution

   1. Step 1: Values from 12 to 20: the classes between those boundaries$12 < h \le 20$: the classes $12 < h \le 16$ to $16 < h \le 20$.
   2. Step 2: Add the frequencies of those classes$3 + 15 = 18$18 of the seedlings have $12 < h \le 20$.
5. Question 5Percentage more than a value

   The histogram shows the distances, $d$ km, that the members of a cycling club rode last Sunday. Find the percentage of the members with $d \le 30$, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Values at most 30: the classes up to the boundary 30$d \le 30$: the classes $20 < d \le 30$ to $20 < d \le 30$.
   2. Step 2: Add the frequencies of those classes$1 = 1$
   3. Step 3: Total frequency: add the heights of all the barsTotal $= 1 + 11 + 9 + 10 + 7$$= 38$
   4. Step 4: Percentage = the number counted ÷ the total × 100%Percentage $= \dfrac{1}{38} \times 100\%$$= 2.6315\ldots\%$$= 2.63\%$ (3 s.f.)
6. Question 6Complete the frequency table

   The histogram shows the marks, $x$, of some pupils in a test. Complete the frequency table.

   Show the worked solution

   1. Step 1: Each missing frequency is the height of that class's bar$30 < x \le 40$: the bar's height is $10$.$70 < x \le 80$: the bar's height is $14$.$80 < x \le 90$: the bar's height is $13$.
7. Question 7Estimate the mean from mid-values

   The histogram shows the heights, $h$ cm, of some seedlings after two weeks. Using the mid-value of each class, calculate an estimate of the mean, correct to 3 significant figures.

   Show the worked solution

   1. Step 1: Read the height of each bar on the frequency axisFrequencies: $13,\ 12,\ 14,\ 1,\ 9$
   2. Step 2: Mid-value of each class: halfway between its boundariesMid-values: $2,\ 6,\ 10,\ 14,\ 18$
   3. Step 3: Total frequencyTotal frequency $= 13 + 12 + 14 + 1 + 9$$= 49$
   4. Step 4: Estimated total: each mid-value × its frequency, addedTotal $= 2 \times 13 + 6 \times 12 + 10 \times 14 + 14 \times 1 + 18 \times 9$$= 414$
   5. Step 5: Estimated mean = total ÷ total frequencyMean $= \dfrac{414}{49}$$= 8.4489\ldots$$= 8.45$ cm (3 s.f.)