CIE IG MATH: STATISTICS

IGCSE MATH: STATISTICS

Histograms with Unequal Class Widths

Time (t mins)	Frequency	Class Width	Frequency Density (Frequency ÷ Width)
0 ≤ t < 10	20	10	2.0
10 ≤ t < 20	30	10	3.0
20 ≤ t < 40	50	20	2.5
40 ≤ t < 60	30	20	1.5

Important: For unequal class widths, plot frequency density on y-axis, not frequency.

3. Measures of Average for Individual and Discrete Data

Mean (x̄)

x̄ = Σx ÷ n

Example: 12, 15, 18, 20, 25

Sum = 12+15+18+20+25 = 90

n = 5

Mean = 90 ÷ 5 = 18

✓ Uses all values

✗ Affected by outliers

✓ Can be decimal

Median

Middle value when ordered

Odd n: Position = (n+1)/2

Even n: Average of two middle values

Example (even): 4, 6, 8, 10, 12, 14

n=6, positions 3 & 4: 8 & 10

Median = (8+10)÷2 = 9

✓ Not affected by outliers

✓ Always from dataset

Mode

Most frequent value

Example: 2, 3, 3, 3, 5, 7, 7, 9

Mode = 3 (appears 3 times)

✓ Easy to find

✓ Works with categorical data

✓ Not affected by outliers

Range (Measure of Spread)

Range = Highest value - Lowest value

Example: 12, 15, 18, 20, 25

Range = 25 - 12 = 13

Note: Range is NOT an average - it measures spread.

Which Average to Use?

Situation	Best Average	Reason
Normal distribution, no outliers	Mean	Uses all data effectively
Data has extreme values	Median	Not affected by outliers
Categorical data	Mode	Only average that works
Finding most popular item	Mode	Shows highest frequency

4. Grouped Continuous Data

Frequency Tables with Midpoints

Height (h cm)	Frequency (f)	Midpoint (x)	f × x
140 ≤ h < 150	8	145	1160
150 ≤ h < 160	15	155	2325
160 ≤ h < 170	12	165	1980
170 ≤ h < 180	5	175	875
Total	40	-	6340

Estimated Mean for Grouped Data

x̄ = Σ(f × x) ÷ Σf

Where: f = frequency, x = midpoint, Σf = total frequency

From table: Mean = 6340 ÷ 40 = 158.5 cm

Note: This is an estimate because we use midpoints.

Modal Class

Class interval with the highest frequency.

Example: In table above, modal class = 150 ≤ h < 160 (f=15)

Median from Grouped Data

Find median position: (n+1)÷2 = (40+1)÷2 = 20.5

Add cumulative frequencies to find which class contains the median.

Median class = class containing the 20.5th value.

5. Cumulative Frequency Diagrams

Cumulative Frequency Table

Time (t mins)	Frequency	Cumulative Frequency
0 ≤ t < 10	8	8
10 ≤ t < 20	15	23
20 ≤ t < 30	22	45
30 ≤ t < 40	12	57
40 ≤ t < 50	3	60

Median from Curve

Position = n/2 = 60/2 = 30

Draw horizontal line from 30 on y-axis to curve, then down to x-axis.

Quartiles

Lower Quartile (Q₁) = n/4 = 60/4 = 15

Upper Quartile (Q₃) = 3n/4 = 3×60/4 = 45

Read values from curve at these positions.

Interquartile Range (IQR)

IQR = Q₃ - Q₁

Represents the middle 50% of data.

Small IQR = consistent data

Large IQR = spread out data

Using Cumulative Frequency Curves

Example Question: How many students took less than 25 minutes?

Method: Read cumulative frequency at t = 25

If reading = 35, then 35 students took less than 25 minutes.

Example Question: What percentage took more than 35 minutes?

Method: Read cumulative frequency at t = 35 (say, 54)

Number above 35 = 60 - 54 = 6

Percentage = (6/60) × 100% = 10%

6. Correlation and Scatter Diagrams

Positive Correlation

As one variable increases, the other increases.

Example: Height vs Weight

↗ ↗ ↗ ↗ (Upward pattern)

Negative Correlation

As one variable increases, the other decreases.

Example: TV hours vs Test scores

↘ ↘ ↘ ↘ (Downward pattern)

No Correlation

No clear relationship between variables.

Example: Shoe size vs IQ

• • • • (Random scatter)

Describing Correlation

Use two descriptors: strength and direction.

Strength	Description	Example Description
Strong	Points close to a straight line	"Strong positive correlation"
Moderate	Some scatter but clear pattern	"Moderate negative correlation"
Weak	Points widely scattered	"Weak positive correlation"
Perfect	All points on a straight line (rare)	"Perfect positive correlation"

⚠️ Important: Correlation ≠ Causation

Just because two variables are correlated does NOT mean one causes the other.

Example: Ice cream sales and drowning incidents are correlated.

This doesn't mean ice cream causes drowning!

Explanation: Both are caused by a third factor: hot weather.

7. Straight Line of Best Fit

Drawing the Line of Best Fit

Rules:

✓ Use a ruler
✓ Equal number of points above and below the line
✓ Line should pass through the mean point (x̄, ȳ)
✓ Ignore outliers when drawing the line
✓ Extend line slightly beyond data points
✓ Line should follow the trend of the data

Finding the Equation

y = mx + c

Where: m = gradient, c = y-intercept

Finding Gradient (m)

m = (y₂ - y₁) ÷ (x₂ - x₁)

Example: Line passes through (0, 10) and (20, 60)

m = (60 - 10) ÷ (20 - 0) = 50 ÷ 20 = 2.5

Finding y-intercept (c)

Value where line crosses y-axis (when x = 0).

Example: Line crosses at y = 10

c = 10

Equation: y = 2.5x + 10

Interpolation

Estimating values within the data range.

More reliable because you're within known data.

Example: If line gives y = 38 when x = 7 (and data has x values from 0-20).

Extrapolation

Estimating values outside the data range.

Less reliable - trend may not continue.

Example: Estimating y when x = 25 (but data only goes to x = 20).

8. Additional Important Concepts

Box-and-Whisker Plots (Box Plots)

Visual display using five-number summary:

Minimum value
Lower quartile (Q₁)
Median (Q₂)
Upper quartile (Q₃)
Maximum value

Q₁

Q₃

Min

Max

Uses: Compare distributions, identify outliers, show spread.

Outliers using IQR Method

Lower boundary = Q₁ - 1.5 × IQR

Upper boundary = Q₃ + 1.5 × IQR

Any value outside these boundaries is an outlier.

Example: Q₁ = 20, Q₃ = 40, IQR = 20

Lower boundary = 20 - 1.5×20 = -10

Upper boundary = 40 + 1.5×20 = 70

Values < -10 or > 70 are outliers.

Key Formulas Summary

Mean

x̄ = Σx ÷ n

Mean (Grouped)

x̄ = Σ(f×x) ÷ Σf

Median Position

(n+1) ÷ 2

Range

Highest - Lowest

IQR

Q₃ - Q₁

Frequency Density

Frequency ÷ Class Width

Pie Chart Angle

(Frequency ÷ Total) × 360°

Stratified Sampling

(Group ÷ Population) × Sample

9. Exam Tips and Practice Questions

Top Exam Tips

✓ Always Show Working

Even for "show that" questions - markers give method marks.

✓ Use a Ruler

For graphs, lines of best fit, and straight lines.

✓ Label Everything

Axes, scales, units, keys, titles.

✓ Check Pie Charts

Angles should sum to 360°.

✓ Order Data First

Before finding median - this is a common mistake.

✓ Give Context

"The mean is 45 minutes" not just "45".

Common Mistakes to Avoid

✗ Using height instead of frequency density for unequal class widths in histograms
✗ Drawing bar charts with gaps for continuous data
✗ Drawing lines of best fit without a ruler
✗ Confusing correlation with causation
✗ Forgetting to include keys for pictograms and stem-and-leaf
✗ Not checking calculations with estimates

Practice Questions

Question 1: Averages

Find the mean, median, mode, and range of:

12, 15, 15, 18, 20, 22, 28

Question 2: Grouped Data

Calculate an estimate for the mean:

Time (mins)	Frequency
0-10	5
10-20	12
20-30	18
30-40	8
40-50	2

Question 3: Sampling

A school has 450 Year 10s and 350 Year 11s. Calculate a stratified sample of size 40.

Good luck with your IGCSE Statistics! 📊

Remember: Practice makes perfect. Work through lots of past paper questions!