Cumulative frequency The cumulative frequency at a value is the number of data values up to that value. Build it by adding the class frequencies one class at a time (a running total). The table shows the masses of 60 parcels. **Mass, m kg** | **Frequency** | **Mass** | **Cumulative frequency** 0<m <= 2 | 6 | m <= 2 | 6 2<m <= 4 | 14 | m <= 4 | 6+14=20 4<m <= 6 | 20 | m <= 6 | 20+20=40 6<m <= 8 | 12 | m <= 8 | 40+12=52 8<m <= 10 | 8 | m <= 10 | 52+8=60 The last cumulative frequency is the total, 60. From the table, 20 parcels weigh at most 4 kg, and 60-40=20 parcels weigh more than 6 kg. Two kinds of table Read the class intervals before answering. A class written 2<m <= 4 comes from a “less than or equal to” table: its cumulative row counts the values at most each boundary. A class written 2 <= m<4 comes from a “less than” table: its cumulative row counts the values less than each boundary. **Table** | **Class looks like** | **Cumulative row gives** | **Cannot give** “Less than or equal to” | 2<m <= 4 | the number with m <= 4 | the number with m<4: values equal to 4 are mixed into the class 2<m <= 4 “Less than” | 2 <= m<4 | the number with m<4 | the number with m <= 4: values equal to 4 are mixed into the class 4 <= m<6 Drawing the cumulative frequency curve - Plot each cumulative frequency at the upper boundary of its class, because every value in the class is at most that boundary: (2,6), (4,20), (6,40), (8,52), (10,60). - Also plot (0,0) at the lower boundary of the first class: no value lies below it. Always start the curve this way. - Join the points with a smooth curve that never goes down. Use the scales given and label both axes. Reading the curve - **A count:** at most 5 kg. Go up from 5 to the curve, then across: about 30 parcels. More than 5 kg: about 60-30=30 parcels. - **A value for a stated fraction:** 80% of the parcels weigh at most m kg. Find 0.8 x 60=48 on the vertical axis, go across to the curve, then down: m is about 7.3. - **A percentage:** the percentage of parcels heavier than 5 kg is about 3060 x 100%=50%. Checks and common errors - Plot at the upper class boundary, not at the mid-value of the class. - The last plotted point is at the total n; the curve never goes down. - Readings from a curve are estimates. Read to about half of one small square. - “More than” and “at least” counts are found by subtracting a reading from the total. - A curve drawn from a “less than or equal to” table answers “at most” questions; it cannot give the exact number below a class boundary.

Cumulative frequency tables and curves

Secondary 4

Cumulative frequency

The cumulative frequency at a value is the number of data values up to that value. Build it by adding the class frequencies one class at a time (a running total). The table shows the masses of $60$ parcels.

Mass, $m$ kg	Frequency	Mass	Cumulative frequency
$0<m\le 2$	$6$	$m\le 2$	$6$
$2<m\le 4$	$14$	$m\le 4$	$6+14=20$
$4<m\le 6$	$20$	$m\le 6$	$20+20=40$
$6<m\le 8$	$12$	$m\le 8$	$40+12=52$
$8<m\le 10$	$8$	$m\le 10$	$52+8=60$

The last cumulative frequency is the total, $60$. From the table, $20$ parcels weigh at most $4$ kg, and $60-40=20$ parcels weigh more than $6$ kg.

Two kinds of table

Read the class intervals before answering. A class written $2<m\le 4$ comes from a “less than or equal to” table: its cumulative row counts the values at most each boundary. A class written $2\le m<4$ comes from a “less than” table: its cumulative row counts the values less than each boundary.

Table	Class looks like	Cumulative row gives	Cannot give
“Less than or equal to”	$2<m\le 4$	the number with $m\le 4$	the number with $m<4$: values equal to $4$ are mixed into the class $2<m\le 4$
“Less than”	$2\le m<4$	the number with $m<4$	the number with $m\le 4$: values equal to $4$ are mixed into the class $4\le m<6$

Drawing the cumulative frequency curve

Plot each cumulative frequency at the upper boundary of its class, because every value in the class is at most that boundary: $(2,6)$, $(4,20)$, $(6,40)$, $(8,52)$, $(10,60)$.
Also plot $(0,0)$ at the lower boundary of the first class: no value lies below it. Always start the curve this way.
Join the points with a smooth curve that never goes down. Use the scales given and label both axes.

Reading the curve

A count: at most $5$ kg. Go up from $5$ to the curve, then across: about $30$ parcels. More than $5$ kg: about $60-30=30$ parcels.
A value for a stated fraction: $80\%$ of the parcels weigh at most $m$ kg. Find $0.8\times60=48$ on the vertical axis, go across to the curve, then down: $m$ is about $7.3$.
A percentage: the percentage of parcels heavier than $5$ kg is about $\dfrac{30}{60}\times100\%=50\%$.

Checks and common errors

Plot at the upper class boundary, not at the mid-value of the class.
The last plotted point is at the total $n$; the curve never goes down.
Readings from a curve are estimates. Read to about half of one small square.
“More than” and “at least” counts are found by subtracting a reading from the total.
A curve drawn from a “less than or equal to” table answers “at most” questions; it cannot give the exact number below a class boundary.

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Mass, \(m\) kg	Frequency	Mass	Cumulative frequency
\(0<m\le 2\)	\(6\)	\(m\le 2\)	\(6\)
\(2<m\le 4\)	\(14\)	\(m\le 4\)	\(6+14=20\)
\(4<m\le 6\)	\(20\)	\(m\le 6\)	\(20+20=40\)
\(6<m\le 8\)	\(12\)	\(m\le 8\)	\(40+12=52\)
\(8<m\le 10\)	\(8\)	\(m\le 10\)	\(52+8=60\)

Table	Class looks like	Cumulative row gives	Cannot give
“Less than or equal to”	\(2<m\le 4\)	the number with \(m\le 4\)	the number with \(m<4\): values equal to \(4\) are mixed into the class \(2<m\le 4\)
“Less than”	\(2\le m<4\)	the number with \(m<4\)	the number with \(m\le 4\): values equal to \(4\) are mixed into the class \(4\le m<6\)