IB Math AA SL Topic 4.3: Central tendency & spread | Notes & Flashcards | Aimnova

IB Math AA SL — Central tendency & spread

Topic 4.3 of IB Mathematics: Analysis and Approaches covers Central tendency & spread, which is part of Unit 4: Statistics & Probability. Students explore key concepts including Mean, median & mode, Grouped data, Quartiles & IQR, Standard deviation. A strong understanding of central tendency & spread is essential for IB Math AA SL exams and builds the foundation for connected topics across the syllabus.

Key concepts in Central tendency & spread

Key Idea: This topic is about summarising a data set with a few numbers — an average (centre) and a spread. It runs through both papers: by-hand work on Paper 1 and 1-Var Stats on the GDC for Paper 2.

📊 The three averages

\bar{x} = \frac{\sum f x}{\sum f}

$x$: each value (or class **midpoint** for grouped data)
$f$: the frequency of that value
$\sum f$: the total frequency = how many data items

Multiply each value by its frequency, add up Σfx, then divide by the total frequency Σf — not by the number of rows. The mean also gives the total: sum = mean × n, which is how you find a missing value.

📦 Spread: range, IQR & standard deviation

\text{range} = \max - \min, \qquad \text{IQR} = Q_3 - Q_1

$Q_1$: lower quartile — median of the lower half
$Q_3$: upper quartile — median of the upper half
$\sigma$: standard deviation — typical distance from the mean (variance = σ²)

Add c to every value → mean + c, σ unchanged (the spread doesn't move). Multiply by k → mean and σ both × |k|. On the GDC read σx (population SD — the IB syllabus value), never Sx. If one value is far from the rest, quote the median, which resists outliers.

✏️ IB-style worked examples

IB-style question — mean from a frequency table (4.3.1)

The number of pets owned by 30 students is recorded: 0 → 11, 1 → 9, 2 → 7, 3 → 3. Find the mean number of pets.

Step by step:

Form Σfx — multiply each value by its frequency and add.
$0(11)+1(9)+2(7)+3(3) = 0+9+14+9 = 32$
Divide by the total frequency Σf = 30.
$\bar{x} = \frac{32}{30} \approx 1.07$

Final answer:

Mean ≈ 1.07 pets.

IB-style question — estimated mean of grouped data (4.3.2)

Journey times (minutes) for 40 commuters are grouped: 0–10 → 6, 10–20 → 14, 20–30 → 12, 30–40 → 8. Estimate the mean and state the modal class.

Step by step:

Use class midpoints 5, 15, 25, 35 as the values, then form Σfx.
$5(6)+15(14)+25(12)+35(8) = 30+210+300+280 = 820$
Divide by Σf = 40 for the estimated mean.
$\bar{x} \approx \frac{820}{40} = 20.5$
The modal class has the greatest frequency (14).
$10 \le t < 20$

Final answer:

Estimated mean ≈ 20.5 min; modal class 10 ≤ t < 20.

IB-style question — median, quartiles & IQR (4.3.3)

Find the median, Q₁, Q₃ and the IQR of 6, 9, 9, 11, 14, 17, 21.

Step by step:

n = 7 (already ordered); the median is the 4th value.
$\text{median} = 11$
Leave the median out: lower half 6, 9, 9 and upper half 14, 17, 21.
$Q_1 = 9, \quad Q_3 = 17$
IQR is the spread of the middle 50%.
$\text{IQR} = 17 - 9 = 8$

Final answer:

Median = 11, Q₁ = 9, Q₃ = 17, IQR = 8.

IB-style question — shift and scale the data (4.3.4)

A data set has mean 30 and standard deviation 6. Every value is decreased by 4, then the result is multiplied by 3. Find the new mean and standard deviation.

Step by step:

Subtract 4: the mean drops by 4, the spread (σ) is unchanged.
$\text{mean} = 26, \quad \sigma = 6$
Multiply by 3: both the mean and σ are multiplied by |3|.
$\text{mean} = 78, \quad \sigma = 18$

Final answer:

New mean = 78, new standard deviation = 18.

Important: For Σfx ÷ Σf, the denominator is the total frequency (add up the f-column), not the number of different values. In the pets example divide by 30, not 4. The same trap hits grouped data — divide by Σf, and remember the answer is only an estimate because you used midpoints.

Tap each card to reveal the answer.

Exam Tips

Order the data before finding the median or quartiles; for odd n, leave the median out of both halves.
Mean from a table = Σfx ÷ Σf — divide by the TOTAL frequency, not the number of rows.
Grouped data: use class midpoints; the mean is an ESTIMATE because exact values are lost.
Adding c leaves σ unchanged; multiplying by k multiplies the mean and σ by |k|.
On Paper 2, one 1-Var Stats run gives x̄, σx and Q₁/Med/Q₃ — read σx, not Sx.

IB Math AA SL — Central tendency & spread

Key concepts in Central tendency & spread

Key Idea: This topic is about summarising a data set with a few numbers — an average (centre) and a spread. It runs through both papers: by-hand work on Paper 1 and 1-Var Stats on the GDC for Paper 2.

📊 The three averages

\bar{x} = \frac{\sum f x}{\sum f}

$x$: each value (or class **midpoint** for grouped data)
$f$: the frequency of that value
$\sum f$: the total frequency = how many data items

Multiply each value by its frequency, add up Σfx, then divide by the total frequency Σf — not by the number of rows. The mean also gives the total: sum = mean × n, which is how you find a missing value.

📦 Spread: range, IQR & standard deviation

\text{range} = \max - \min, \qquad \text{IQR} = Q_3 - Q_1

$Q_1$: lower quartile — median of the lower half
$Q_3$: upper quartile — median of the upper half
$\sigma$: standard deviation — typical distance from the mean (variance = σ²)

Add c to every value → mean + c, σ unchanged (the spread doesn't move). Multiply by k → mean and σ both × |k|. On the GDC read σx (population SD — the IB syllabus value), never Sx. If one value is far from the rest, quote the median, which resists outliers.

✏️ IB-style worked examples

IB-style question — mean from a frequency table (4.3.1)

The number of pets owned by 30 students is recorded: 0 → 11, 1 → 9, 2 → 7, 3 → 3. Find the mean number of pets.

Step by step:

Form Σfx — multiply each value by its frequency and add.
$0(11)+1(9)+2(7)+3(3) = 0+9+14+9 = 32$
Divide by the total frequency Σf = 30.
$\bar{x} = \frac{32}{30} \approx 1.07$

Final answer:

Mean ≈ 1.07 pets.

IB-style question — estimated mean of grouped data (4.3.2)

Journey times (minutes) for 40 commuters are grouped: 0–10 → 6, 10–20 → 14, 20–30 → 12, 30–40 → 8. Estimate the mean and state the modal class.

Step by step:

Use class midpoints 5, 15, 25, 35 as the values, then form Σfx.
$5(6)+15(14)+25(12)+35(8) = 30+210+300+280 = 820$
Divide by Σf = 40 for the estimated mean.
$\bar{x} \approx \frac{820}{40} = 20.5$
The modal class has the greatest frequency (14).
$10 \le t < 20$

Final answer:

Estimated mean ≈ 20.5 min; modal class 10 ≤ t < 20.

IB-style question — median, quartiles & IQR (4.3.3)

Find the median, Q₁, Q₃ and the IQR of 6, 9, 9, 11, 14, 17, 21.

Step by step:

n = 7 (already ordered); the median is the 4th value.
$\text{median} = 11$
Leave the median out: lower half 6, 9, 9 and upper half 14, 17, 21.
$Q_1 = 9, \quad Q_3 = 17$
IQR is the spread of the middle 50%.
$\text{IQR} = 17 - 9 = 8$

Final answer:

Median = 11, Q₁ = 9, Q₃ = 17, IQR = 8.

IB-style question — shift and scale the data (4.3.4)

A data set has mean 30 and standard deviation 6. Every value is decreased by 4, then the result is multiplied by 3. Find the new mean and standard deviation.

Step by step:

Subtract 4: the mean drops by 4, the spread (σ) is unchanged.
$\text{mean} = 26, \quad \sigma = 6$
Multiply by 3: both the mean and σ are multiplied by |3|.
$\text{mean} = 78, \quad \sigma = 18$

Final answer:

New mean = 78, new standard deviation = 18.

Important: For Σfx ÷ Σf, the denominator is the total frequency (add up the f-column), not the number of different values. In the pets example divide by 30, not 4. The same trap hits grouped data — divide by Σf, and remember the answer is only an estimate because you used midpoints.

Tap each card to reveal the answer.

Exam Tips

Order the data before finding the median or quartiles; for odd n, leave the median out of both halves.
Mean from a table = Σfx ÷ Σf — divide by the TOTAL frequency, not the number of rows.
Grouped data: use class midpoints; the mean is an ESTIMATE because exact values are lost.
Adding c leaves σ unchanged; multiplying by k multiplies the mean and σ by |k|.
On Paper 2, one 1-Var Stats run gives x̄, σx and Q₁/Med/Q₃ — read σx, not Sx.

IB Math AA SL — Central tendency & spread

Key concepts in Central tendency & spread

📊 The three averages

📦 Spread: range, IQR & standard deviation

✏️ IB-style worked examples

IB-style question — mean from a frequency table (4.3.1)

IB-style question — estimated mean of grouped data (4.3.2)

IB-style question — median, quartiles & IQR (4.3.3)

IB-style question — shift and scale the data (4.3.4)

Exam Tips

What you'll learn in Topic 4.3

Study resources — 4.3 Central tendency & spread

Mean, median & mode

Grouped data

Quartiles & IQR

Standard deviation

Ready to study Central tendency & spread?

Ready to practice?

IB Math AA SL — Central tendency & spread

Key concepts in Central tendency & spread

📊 The three averages

📦 Spread: range, IQR & standard deviation

✏️ IB-style worked examples

IB-style question — mean from a frequency table (4.3.1)

IB-style question — estimated mean of grouped data (4.3.2)

IB-style question — median, quartiles & IQR (4.3.3)

IB-style question — shift and scale the data (4.3.4)

Exam Tips

What you'll learn in Topic 4.3

Study resources — 4.3 Central tendency & spread

Mean, median & mode

Grouped data

Quartiles & IQR

Standard deviation

Ready to study Central tendency & spread?

Ready to practice?