Drawing Cumulative Frequency Diagrams

Worked example

1. 1

   $f:6,14,20,20,14,6$ — The question: times of 80 people, in six classes of 10 minutes from 0 to 60.

2. 2

   $\text{CF}:6,20,40,60,74,80$ — First make the running totals: add each frequency to the total so far. The last one is 80, everyone.

3. 3

   $\text{start at }(0,0)$ — The first class starts at 0 and nobody took less than that, so the graph starts at zero people.

4. 4

   $(10,6)$ — By the upper end of the first class, 10 minutes, all 6 people in it are counted. So plot 6 above 10.

5. 5

   $(20,20)$ — Same rule: upper end of the second class, with its running total.

6. 6

   $(30,40)$ — Upper end 30, running total 40.

7. 7

   $(40,60)$ — Upper end 40, running total 60.

8. 8

   $(50,74)$ — Upper end 50, running total 74.

9. 9

   $(60,80)$ — The last point is at the total, 80. Now join all seven points.

Worked example

1. 1

   $\text{up from }45\text{ to the graph}$ — Start on the bottom axis at 45 minutes and go straight up until you hit the curve.

2. 2

   $\text{across: }67$ — Go across to the side axis and read 67. The graph counts people at or below 45, so 67 took 45 minutes or less.

3. 3

   $80-67=13$ — Everyone else took more than 45, so take the 67 away from the total of 80.

Worked example

1. 1

   $\text{at }50:102$ — A graph for 120 runners. Go up from 50 minutes to the curve, then across: 102 runners took 50 minutes or less.

2. 2

   $120-102=18$ — More than 50 is everyone else, so take 102 from the total of 120.

3. 3

   $\frac{18}{120}\times 100=15\%$ — A percentage is the part over the total, times 100.

4. 4

   $P=\frac{18}{120}=\frac{3}{20}$ — A probability is the same fraction, without the times 100. Simplify it: divide top and bottom by 6.

Worked example

1. 1

   $\text{at }35:50$ — Go up from 35 to the curve, then across. 50 people took 35 minutes or less.

2. 2

   $\text{at }25:30$ — Do the same from 25. 30 people took 25 minutes or less.

3. 3

   $50-30=20$ — The 50 already includes the 30 people below 25. Take them away to leave the 20 between 25 and 35.

Worked example

1. 1

   $\frac{1}{2}\times 80=40$ — The median is the middle person, so find half of the 80 people. This is a number of people, on the side axis.

2. 2

   $\text{across at }40\text{, down: }30$ — Go across from 40 to the curve, then down to the time axis. Median = 30 minutes.

3. 3

   $\frac{1}{4}\times 80=20\to 20\text{ min}$ — The lower quartile is a quarter of the way through. Across at 20, down: 20 minutes.

4. 4

   $\frac{3}{4}\times 80=60\to 40\text{ min}$ — The upper quartile is three quarters of the way. Across at 60, down: 40 minutes.

5. 5

   $40-20=20\text{ minutes}$ — Interquartile range = upper quartile − lower quartile. It is the spread of the middle half.