10A

Cumulative frequency and probability

Cumulative frequency and cumulative relative frequency, the cumulative distribution function of a discrete random variable, grouped continuous data, median and quartiles from a cumulative graph, and probability as area under a histogram.

Cumulative frequency

The cumulative frequency of a value is the number of results less than or equal to it. Dividing by the total gives the cumulative relative frequency, which runs from $0$ up to $1$.

Example 1●

$40$ students recorded how many books they read in a month. Complete the table.

Books	Frequency	Cumulative frequency	Cumulative relative frequency
0	6	6	0.15
1	16	22	0.55
2	10	32	0.8
3	8	40	1

Each cumulative frequency adds the next frequency to the one above it. Divide each by $40$ for the last column.

So $80\%$ of students read at most $2$ books.

The cumulative distribution function

CDF of a random variable

$$F(x) = P(X \le x)$$

For a discrete random variable, $F$ is found by adding the probabilities up to and including $x$. Its graph is a step graph that rises from $0$ to $1$ and never decreases.

Example 2: a die●

$X$ is the score on a fair die. Find $F(4)$ and $P(X \gt 4)$.

$F(4) = P(X \le 4) = \tfrac46 = \tfrac23$.

$P(X \gt 4) = 1 - F(4) = \tfrac13$.

Grouped data, median and quartiles

Continuous data such as heights is grouped into classes. On a cumulative relative frequency graph (an ogive), read across from $0.5$ for the median, and from $0.25$ and $0.75$ for the lower and upper quartiles. The mode is the most frequent value, or the modal class for grouped data.

Example 3●●

For the books data in Example 1, find the median and the quartiles.

The cumulative relative frequency jumps from $0.15$ to $0.55$ at $1$ book, so it passes both $0.25$ and $0.5$ there. So $Q_1 = 1$ and the median is $1$.

It first reaches $0.75$ at $2$ books, so $Q_3 = 2$.

Probability as area

In a relative frequency histogram with columns of width $1$, the area of each column equals its relative frequency, so the total area is $1$. The probability of landing in a range is the area over that range. Continuous distributions take this idea to the limit, with a smooth curve in place of the columns.

Flashcards

Cumulative frequency

The number of results less than or equal to a value.

$F(x)$

$P(X \le x)$

Median from an ogive

Read across from cumulative relative frequency $0.5$.

Quartiles from an ogive

Read across from $0.25$ and $0.75$.

Practice

In Example 1, how many students read at most $1$ book?
In Example 1, find the cumulative relative frequency for $2$ books.
In Example 1, what proportion of students read $3$ books?
$X$ is the score on a fair die. Find $F(4)$.
For the die, find $P(X \gt 4)$.
For the die, find $F(6)$.
On an ogive, the median is read from cumulative relative frequency:
The graph of a cumulative distribution function:
In a relative frequency histogram with column width $1$, the total area of the columns is: