Cumulative frequency questions with answers

Questions

1. Question 1Running totals from a table

   The table shows the marks of $50$ students in a test. Find the number of students who scored $30$ marks or fewer.

   Marks, $x$	Frequency
   $0 < x \le 10$	5
   $10 < x \le 20$	11
   $20 < x \le 30$	14
   $30 < x \le 40$	10
   $40 < x \le 50$	6
   $50 < x \le 60$	4

   Show the worked solution

   1. Step 1: A cumulative frequency is a running total: add the frequencies of every class up to that mark$\mathstrut 5 + 11 + 14$$\text{Cumulative frequency} = 30$
2. Question 2Build the cumulative frequency table

   The frequency table shows the time, in minutes, that $50$ students took to finish a puzzle. Complete the cumulative frequency column.

   Show the worked solution

   1. Step 1: The first class: the cumulative frequency is its own frequency$\text{1st} = 5$
   2. Step 2: Each next one: add its frequency to the one above$\mathstrut \text{2nd} = 5 + 11$$\text{2nd} = 16$
   3. Step 3: Carry on down the column. Check: the last one is the total$\text{Last} = 50$Cumulative frequencies: $5, \ 16, \ 30, \ 40, \ 46, \ 50$
3. Question 3Frequencies from cumulative frequencies

   The cumulative frequency table shows the marks of $50$ students in a test. Complete the frequency column.

   Show the worked solution

   1. Step 1: The first class: its frequency is its cumulative frequency$\text{1st} = 4$
   2. Step 2: Each next one: subtract the cumulative frequency above it$\mathstrut \text{2nd} = 13 - 4$$\text{2nd} = 9$
   3. Step 3: Carry on down the column. Check: the frequencies add up to the last cumulative frequency$\text{Total} = 50$Frequencies: $4, \ 9, \ 15, \ 12, \ 7, \ 3$
4. Question 4Read the curve: at most x

   The cumulative frequency curve shows the marks of $40$ students in a test. Use the curve to estimate the number of students who scored at most $15$ marks.

   Show the worked solution

   1. Step 1: Go up from the mark on the horizontal axis to the curve, then acrossThe curve gives about $4$ students with at most $15$ marks.
5. Question 5Read the curve: more than x

   The cumulative frequency curve shows the marks of $60$ students in a test. Use the curve to estimate the number of students who scored more than $45$ marks.

   Show the worked solution

   1. Step 1: Go up from the mark on the horizontal axis to the curve, then acrossThe curve gives about $46$ students with at most $45$ marks.
   2. Step 2: More than: take it from the total$\mathstrut 60 - 46$$\text{Number} \approx 14$