Cumulative Frequency

Cumulative frequency is a running total — the count of values up to and including a given point.

Worked example. Times: $0$–$10$ (5), $10$–$20$ (12), $20$–$30$ (8), $30$–$40$ (5). Total $n = 30$.

Cumulative table:

Time $\le t$	Cumulative freq
$\le 10$	5
$\le 20$	17
$\le 30$	25
$\le 40$	30

KEY RULE. Plot cumulative frequency at the UPPER BOUND of each class — NOT the midpoint, NOT the lower bound.

Why? Because "cumulative frequency at $t = 20$" means "how many had time UP TO 20". The class $10$–$20$ contributes its full 12 to that count.

Edexcel tip. Always start the curve at $(0, 0)$ if there are no values below the first class — otherwise the curve doesn't pass through the origin.

The curve. Plot the cumulative frequencies at the upper bounds, then connect with a SMOOTH CURVE (not straight lines). The shape is an "S" or "ogive".

Median from the curve. Median position is $n/2$.

Worked example. $n = 60$. Median position = $30$.

• Find $30$ on the y-axis.
• Read across to the curve.
• Read down to the x-axis.
• That x-value is the median.

Quartiles from the curve.

Quartile	Position
$Q_1$ (lower quartile)	$n/4$
Median ($Q_2$)	$n/2$
$Q_3$ (upper quartile)	$3n/4$

Worked example. $n = 60$. $Q_1$ at $y = 15$; median at $y = 30$; $Q_3$ at $y = 45$.

Interquartile range (IQR). $\text{IQR} = Q_3 - Q_1$. Measures the spread of the middle 50% — robust to outliers.

Worked example. $Q_1 = 18$, $Q_3 = 32$. IQR $= 14$.

Edexcel tip. Mark schemes give ±1 unit tolerance for graph readings (because precision depends on graph scale). Always show your construction lines on the graph — examiners credit the method.

A box plot displays five numbers:

1. Minimum
2. Lower quartile $Q_1$
3. Median
4. Upper quartile $Q_3$
5. Maximum

Construction.

• Box from $Q_1$ to $Q_3$ with a line at the median.
• Whiskers from min to $Q_1$ and from $Q_3$ to max.

Worked example. $\min = 5$, $Q_1 = 12$, median $= 18$, $Q_3 = 24$, $\max = 40$.

• Box: 12 to 24.
• Median line at 18.
• Left whisker: 5 to 12.
• Right whisker: 24 to 40.

Skewness. Compare the two whisker lengths and the position of the median in the box.

• Right whisker much longer → POSITIVE skew (long tail to the right).
• Left whisker much longer → NEGATIVE skew.
• Symmetric: median in middle of box, whiskers similar length.

Comparing box plots. Edexcel asks "compare the two distributions". TWO statements needed:

• One about CENTRE (compare medians).
• One about SPREAD (compare IQRs).

Worked example. Group A: median 12, IQR 6. Group B: median 18, IQR 4.

• "Group B has a higher median (18 vs 12), so on average Group B values are larger."
• "Group A has a larger IQR (6 vs 4), so Group A values are more spread out."

Edexcel tip. Always state comparisons IN CONTEXT — say "students" or "marks", not just numbers.