Cumulative frequency

The median of a list and of a frequency table, cumulative and relative cumulative frequency, and cumulative frequency histograms and polygons.

The median

Example 1●

Find the median of $3, 7, 8, 12, 15$ and of $2, 4, 7, 9, 10, 13$.

Five values: the 3rd is $8$.

Six values: average the 3rd and 4th, $\dfrac{7 + 9}{2} = 8$.

Cumulative frequency

The cumulative frequency of a score is the number of values less than or equal to it. Running down the table, it climbs to $n$.

Score $x$	5	6	7	8	9	10
Frequency $f$	2	4	7	5	1	1
Cumulative frequency	2	6	13	18	19	20
Relative cumulative	0.1	0.3	0.65	0.9	0.95	1

Example 2●●

Find the median score from the table.

$n = 20$, so the median is the average of the 10th and 11th scores.

Both the 10th and 11th scores are $7$, so the median is $7$.

The cumulative frequency passes 10 at the score 7 (it jumps from 6 to 13).

A cumulative frequency histogram has columns that never go down; the cumulative frequency polygon (ogive) joins the top right corner of each column.

Flashcards

Median position

$\dfrac{n + 1}{2}$

Cumulative frequency

Number of values $\le$ the score.

Median from a table

First score whose cumulative frequency reaches the median position.

Practice

Find the median of $3, 7, 8, 12, 15$.
Find the median of $2, 4, 7, 9, 10, 13$.
For the siblings table in 15A (frequencies $3, 8, 5, 3, 1$ for $0$ to $4$), find the cumulative frequency of $2$.
Find the median number of siblings.
Find the median score from the table above.
For $n = 25$ ordered values, the median is the:
From the table, what fraction of students scored $7$ or less? Give a decimal.
A cumulative frequency histogram:
From the table, how many students scored more than $8$?