What are the measures of location in Statistics 1 for Edexcel International A Levels?

In Statistics 1 for Edexcel International A Levels, measures of location refer to statistical values that describe the central tendency of a data set. The most common measures include the mean, median, and mode. The mean is the average of all data points, the median is the middle value when the data is ordered, and the mode is the most frequently occurring value.

How do you calculate the range as a measure of spread in Edexcel International A Levels Mathematics?

The range is a simple measure of spread that indicates the difference between the highest and lowest values in a data set. To calculate the range, subtract the smallest value from the largest value. This measure gives a quick sense of the data's variability but does not account for any outliers.

What is the importance of standard deviation in understanding data spread in Statistics 1?

Standard deviation is a crucial measure of spread in Statistics 1 as it quantifies the amount of variation or dispersion in a data set. A low standard deviation indicates that data points are close to the mean, while a high standard deviation suggests a wider spread around the mean. This measure is particularly useful for comparing the variability between different data sets.

Can you explain the difference between variance and standard deviation in Edexcel International A Levels Statistics?

Variance and standard deviation both measure the spread of a data set. Variance is the average of the squared differences from the mean, providing a measure of how data points differ from the mean. Standard deviation is the square root of the variance, offering a more interpretable measure of spread in the same units as the data. Both are essential in Edexcel International A Levels Statistics for understanding data variability.

Why is the interquartile range (IQR) preferred over the range in some cases in Statistics 1?

The interquartile range (IQR) is often preferred over the range because it provides a measure of spread that is not affected by outliers. The IQR is the difference between the third quartile (Q3) and the first quartile (Q1), representing the middle 50% of the data. This makes it a more robust measure of spread, especially in skewed distributions or when outliers are present, which is a key consideration in Edexcel International A Levels Statistics.

Why is "Using class boundaries instead of midpoints in $\sum fx$" a common mistake in Measures of location and spread?

Why it happens: Students copy the boundary from the table without computing the midpoint. How to avoid it: For each class [a, b), the midpoint is (a + b)/2. Write the midpoint as a separate column before computing fx. Source: WST01/01 examiner reports — annual

Why is "Dividing by $n - 1$ for S1 standard deviation" a common mistake in Measures of location and spread?

Why it happens: Calculator default setting or memory of statistics from another course. How to avoid it: S1 uses the population formula ^2 = x^2/n - x^2. Check your calculator's \sigma_n vs \sigma_n-1 button. The formula booklet uses n.

Why is "Claiming the SD changes when a constant is added" a common mistake in Measures of location and spread?

Why it happens: Confusion between shift and scale. How to avoid it: If y = x + b: SD is unchanged. If y = ax: SD multiplies by |a|. If y = ax + b: SD multiplies by |a| only — the b has no effect. Source: WST01/01 examiner reports — recurring

Why is "Using $(n + 1)/4$ for $Q_1$ on grouped data" a common mistake in Measures of location and spread?

Why it happens: The (n+1)/4 rule is correct for raw discrete data but not for the Edexcel-S1 grouped-data convention. How to avoid it: For grouped/continuous data, Edexcel uses n/4, n/2, 3n/4 with linear interpolation. For raw data, use (n+1)/4. The question wording usually makes the data type clear.

Why is "Omitting units from the final mean or SD" a common mistake in Measures of location and spread?

Why it happens: Students focus on the number. How to avoid it: Always quote the unit (kg, min, marks, m). A-marks for the final answer often depend on units being present.

Why is "Saying 'A has bigger SD than B' without context" a common mistake in Measures of location and spread?

Why it happens: Generic statistical phrasing. How to avoid it: Always finish a comparison with what it means in context: 'Class A is more consistent in mark — typical deviation 8 marks vs 22 in Class B.' Examiners look for the named variable in the sentence.

Edexcel International A Levels Mathematics (XMA01-YMA01)

Measures of location and spread

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Short Notes - Measures of location and spread

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Detailed Study Notes

Detailed notes on Statistics 1 for Edexcel International A Levels Mathematics, covering key concepts, explanations, examples, and exam-focused revision points.

Measures of Location and Spread Study Notes — Edexcel IAL Mathematics S1 (WST01/01, 2018 spec — 2026 onwards)

All the summary statistics you need for S1: mean, median, mode for centre; range, IQR, variance, SD for spread; coding to simplify arithmetic; and the discipline of comparing two data sets with the right statistics for the right situation. The workhorse chapter of S1.

At a glance

Mean $\bar{x} = \sum fx / \sum f$ — sensitive to outliers.
Median — middle value, robust to outliers; for grouped data use linear interpolation at $n/2$ .
Quartiles at $n/4$ , $n/2$ , $3n/4$ (grouped) or $(n+1)/4$ , $(n+1)/2$ , $3(n+1)/4$ (raw).
IQR $= Q_3 - Q_1$ — robust spread.
Variance $\sigma^{2} = \sum x^{2}/n - \bar{x}^{2}$ (booklet shortcut).
SD $\sigma = \sqrt{\sigma^{2}}$ , same units as $x$ .
Coding $y = ax + b$ : $\bar{y} = a\bar{x} + b$ , $\sigma_y = |a|\sigma_x$ .
Comparing two data sets: outliers $\Rightarrow$ median + IQR; symmetric $\Rightarrow$ mean + SD.

What you’ll learn

Mapped to the Pearson Edexcel International A Levels XMA01-YMA01 syllabus (2026 onwards).

S1 2.1 — Calculate measures of location: mean, median, mode for raw, frequency-tabulated and grouped data.
S1 2.2 — Calculate measures of spread: range, interquartile range, variance and standard deviation.
S1 2.3 — Use the coding $y = (x - a)/h$ and recover the mean and standard deviation of the original data.
S1 2.4 — Use linear interpolation to estimate the median and quartiles from grouped data.
S1 2.5 — Compare two data sets using appropriate measures of location and spread, justified by the context.

Measures of centre — mean, median, mode

Three different answers to the same question 'what is typical?'.

Mean. Arithmetic average.

$\bar{x} = \dfrac{\sum x}{n}, \qquad \bar{x} = \dfrac{\sum fx}{\sum f}.$

For grouped data, $x$ is the midpoint of each class.

Strengths: uses every value, mathematically tractable, links to variance.
Weaknesses: pulled by outliers; can be misleading when data is skewed.

Median. Middle value of the ordered data.

Raw data of size $n$ : median = $((n+1)/2)$ -th value (interpolating between two values when $n$ is even).
Grouped data: locate position $n/2$ in the cumulative frequency, then linear-interpolate within the median class.

Strengths: robust to outliers; meaningful for skewed data. Weakness: ignores most of the data.

Mode. Most frequent value (or modal class for grouped data).

Strength: works for categorical/qualitative data.
Weakness: may not exist or may be multiple. Least informative for continuous data.

Choosing. Symmetric, no outliers $\Rightarrow$ mean. Skewed or outliers present $\Rightarrow$ median. Categorical $\Rightarrow$ mode.

Symmetric data: mean = median = mode. Right-skewed: mean > median > mode. Left-skewed: mean < median < mode.

Mean uses all data — sensitive to outliers.
Median = middle — robust.
Mode = most frequent — best for categorical data.
For grouped data, mean uses midpoints; median uses linear interpolation.

Quartiles, percentiles and linear interpolation

Position by $n/4, n/2, 3n/4$ ; then interpolate inside the class.

Quartile positions for grouped data.

Quartile	Position	What it means
$Q_1$	$n/4$	$25\%$ of data below
$Q_2$ (median)	$n/2$	$50\%$ of data below
$Q_3$	$3n/4$	$75\%$ of data below

Linear interpolation. Inside the relevant class with lower boundary $L$ , width $w$ , cumulative frequency $C_{\text{before}}$ before the class and $f$ within it, the quartile at position $p$ is

$Q = L + \dfrac{p - C_{\text{before}}}{f} \cdot w.$

Worked example. From cumulative frequencies $4, 12, 24, 38, 44, 48, 50$ at marks $\le 20, 30, 40, 50, 60, 70, 80$ :

$n = 50$ , $n/4 = 12.5$ . Class $30 < x \le 40$ (CF $12 \to 24$ ): $Q_1 = 30 + (12.5 - 12)/12 \cdot 10 \approx 30.42$ .
$n/2 = 25$ . Class $40 < x \le 50$ : $Q_2 = 40 + (25 - 24)/14 \cdot 10 \approx 40.71$ .
$3n/4 = 37.5$ . Same class: $Q_3 = 40 + (37.5 - 24)/14 \cdot 10 \approx 49.64$ .

So IQR $\approx 19.2$ .

Percentiles. $P_k$ is the value below which $k\%$ of the data lies. Position is $kn/100$ for grouped data; same interpolation method.

Positions $n/4$ , $n/2$ , $3n/4$ for grouped data.
Linear interpolation: $Q = L + (p - C_{\text{before}})/f \cdot w$ .
Percentiles use the same approach with position $kn/100$ .
For raw discrete data, use $(n+1)/4$ etc.

Variance and standard deviation

$\sigma^{2}$ = mean of squares minus square of mean. SD = $\sqrt{}$ .

Definitions. Variance is the mean squared deviation from the mean.

$\sigma^{2} = \dfrac{\sum (x - \bar{x})^{2}}{n}.$

Shortcut formula (in the booklet). Algebraic manipulation gives

$\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}, \qquad \sigma^{2} = \dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}.$

Always use the shortcut for calculator-friendly arithmetic.

Standard deviation $\sigma = \sqrt{\sigma^{2}}$ — same units as the data.

Worked example. $n = 20$ , $\sum x = 850$ , $\sum x^{2} = 36{,}450$ .

$\bar{x} = 850/20 = 42.5$ .
$\sigma^{2} = 36{,}450/20 - 42.5^{2} = 1822.5 - 1806.25 = 16.25$ .
$\sigma = \sqrt{16.25} \approx 4.03$ .

Population vs sample. S1 uses the population formula (divide by $n$ ). The sample variance $s^{2} = \sum(x - \bar{x})^{2}/(n-1)$ appears in higher-level statistics but not in S1 — your calculator's $\sigma_n$ button gives the S1 answer.

Interpretation. $\sigma$ is the typical distance of a data point from the mean. For approximately normal data, $\approx 68\%$ of values lie within $\bar{x} \pm \sigma$ , $\approx 95\%$ within $\bar{x} \pm 2\sigma$ .

Definition: $\sigma^{2}$ = mean of squared deviations from mean.
Shortcut: $\sigma^{2} = \overline{x^{2}} - \bar{x}^{2}$ — IS in the booklet.
S1 always divides by $n$ , never $n-1$ .
SD has same units as $x$ .

Coding — making arithmetic easy

Replace big/awkward numbers by a linear $y = (x - a)/h$ . Mean shifts and scales; SD only scales.

Why code? When data values are large or have an awkward common shift (e.g. heights all around $175$ cm), define

$y = \dfrac{x - a}{h}.$

Compute $\bar{y}$ and $\sigma_y$ on the simpler $y$ values, then back-transform.

Back-transformation rules. From $x = hy + a$ :

$\bar{x} = h\bar{y} + a, \qquad \sigma_x = |h|\,\sigma_y.$

Worked example. $y = (x - 200)/5$ gives $\bar{y} = 4$ , $\sigma_y = 2$ .

$\bar{x} = 5 \times 4 + 200 = 220$ .
$\sigma_x = 5 \times 2 = 10$ .

Key fact. Shifts do not change spread. Adding $a$ to every value moves the mean by $a$ but leaves $\sigma$ unchanged. Only the scale factor $h$ affects $\sigma$ .

Common application. Calculator-friendly coding for a table of heights: $y = x - 170$ . The $y$ -values are around $0$ , easy to square and sum, then $\bar{x} = \bar{y} + 170$ and $\sigma_x = \sigma_y$ .

$y = (x - a)/h$ rearranges to $x = hy + a$ .
Mean: $\bar{x} = h\bar{y} + a$ .
SD: $\sigma_x = |h|\sigma_y$ — shift doesn't matter.
Variance: $\sigma_x^{2} = h^{2}\sigma_y^{2}$ .

Comparing two data sets

Centre + spread, in context, using statistics suited to the data shape.

Step 1 — choose statistics.

Situation	Centre	Spread
Roughly symmetric, no outliers	mean	SD
Skewed or outliers present	median	IQR

If one data set has outliers but the other doesn't, use the median + IQR for both so the comparison is on the same footing.

Step 2 — write two sentences.

One sentence on centre (in context): 'Class A has a higher mean mark ( $64$ ) than Class B's median ( $62$ ), so typical performance is similar.'
One sentence on spread (in context): 'Class B is much more variable (IQR $= 22$ ) than Class A (SD $= 8$ ), so Class A is more consistent.'

Step 3 — add interpretation. Use the variable name from the question (marks, heights, times). Examiners reward 'Class B has a wider range of mark', NOT 'Class B has a higher value'.

Pitfall. Comparing mean of one set against median of the other only works if both sets are reasonably symmetric so that mean $\approx$ median. Otherwise re-quote both statistics on the same scale.

Outliers $\Rightarrow$ median + IQR; symmetric $\Rightarrow$ mean + SD.
Use the same pair of statistics for both data sets.
Write two sentences: centre and spread.
Always cite the actual variable (marks, times) — context wins marks.

Quick recap

Mean = $\sum fx / \sum f$ (use midpoints for grouped).
Median = $(n/2)$ -th by linear interpolation (grouped) / $((n+1)/2)$ -th (raw).
Quartiles at $n/4$ , $n/2$ , $3n/4$ by linear interpolation.
IQR = $Q_3 - Q_1$ (robust spread).
Variance $\sigma^{2} = \sum x^{2}/n - \bar{x}^{2}$ — divide by $n$ .
Coding $y = (x - a)/h$ : $\bar{x} = h\bar{y} + a$ , $\sigma_x = |h|\sigma_y$ .
Compare two sets: outliers $\to$ median/IQR, no outliers $\to$ mean/SD, in context.

Memorise this

Verbatim phrases and definitions Edexcel mark schemes credit.

Mean (frequency): $\bar{x} = \sum fx / \sum f$ .
Variance shortcut: $\sigma^{2} = \sum x^{2}/n - \bar{x}^{2}$ .
SD: $\sigma = \sqrt{\sigma^{2}}$ .
Quartile positions (grouped): $n/4$ , $n/2$ , $3n/4$ .
Linear interpolation: $Q = L + \dfrac{p - C_{\text{before}}}{f} \cdot w$ .
Coding: $y = (x - a)/h \Rightarrow \bar{x} = h\bar{y} + a$ , $\sigma_x = |h|\sigma_y$ .

How it’s examined

Measures of location and spread appears in every WST01/01 paper (Jan & Jun), typically Q3 or Q4 worth $7$ - $10$ marks total. Typical tasks: estimate the mean from a grouped table (3-4 marks); use the booklet shortcut to find SD from $\sum x$ , $\sum x^{2}$ (3-4 marks); apply coding to recover the original mean / SD (4-5 marks); locate quartiles by linear interpolation (5-6 marks); compare two data sets giving reasons for the choice of statistics (4-5 marks). Mark schemes require correctly-cited formulae, substitution shown, and a contextual sentence in any comparison.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WST01/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Measures of location and spread

Step-by-step solutions to past-paper-style questions on measures of location and spread, written exactly the way a tutor would explain them at the board.

1Mean from a grouped frequency table (4 marks, S1)
Core• Adapted from WST01/01 January 2024 Q3• grouped data, mean
Question
The table shows the time, $t$ minutes, taken by $40$ students to complete a homework task. Estimate the mean time.

$t$ (mins) $0 \le t < 10$ $10 \le t < 20$ $20 \le t < 30$ $30 \le t < 40$
frequency $f$ $4$ $12$ $16$ $8$
Step-by-step solution
1. Step 1
  For grouped data use the midpoints. Midpoints are $5, 15, 25, 35$ .
2. Step 2
  Compute $\sum fx$ using midpoint $\times$ frequency:
  $\sum fx = 4(5) + 12(15) + 16(25) + 8(35) = 20 + 180 + 400 + 280 = 880$
3. Step 3
  Total frequency: $\sum f = 4 + 12 + 16 + 8 = 40$ .
4. Step 4
  Apply the formula $\bar{t} = \dfrac{\sum fx}{\sum f}$ :
  $\bar{t} = \dfrac{880}{40} = 22\,\text{min}$
Answer
$\bar{t} = 22$ minutes (estimate from grouped data).
Examiner tip
M1 for using midpoints. M1 for $\sum fx$ correct. A1 for $880$ . A1 for the final answer with units. Mark schemes accept $22.0$ . Examiners deduct if the answer is given without units or if the candidate uses class boundaries instead of midpoints.
2Variance and SD using the shortcut formula (5 marks, S1)
Extended• Adapted from WST01/01 June 2024 Q4• variance, standard deviation, shortcut formula
Question
The mass, $x$ kg, of $20$ children is summarised by $\sum x = 850$ and $\sum x^{2} = 36{,}450$ . Find the mean and the standard deviation of the masses.
Step-by-step solution
1. Step 1
  Mean:
  $\bar{x} = \dfrac{\sum x}{n} = \dfrac{850}{20} = 42.5\,\text{kg}$
2. Step 2
  Apply the shortcut formula for variance from the formula booklet:
  $\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}$
3. Step 3
  Substitute:
  $\sigma^{2} = \dfrac{36{,}450}{20} - 42.5^{2} = 1822.5 - 1806.25 = 16.25$
4. Step 4
  Standard deviation:
  $\sigma = \sqrt{16.25} \approx 4.03\,\text{kg}$
Answer
$\bar{x} = 42.5$ kg, $\sigma \approx 4.03$ kg (3 sf).
Examiner tip
B1 for mean. M1 for substituting into the booklet variance shortcut. A1 for $\sigma^{2} = 16.25$ . A1 for $\sigma$ . The booklet formula is $\sigma^{2} = \sum x^{2}/n - \bar{x}^{2}$ — using $n - 1$ instead of $n$ here is wrong for the population SD that S1 expects.
3Quartiles by linear interpolation (6 marks, S1)
Extended• Adapted from WST01/01 January 2023 Q4• quartiles, interpolation, grouped data
Question
The cumulative frequency table shows the marks of $50$ students.

Mark $\le$ $20$ $30$ $40$ $50$ $60$ $70$ $80$
Cumulative frequency $4$ $12$ $24$ $38$ $44$ $48$ $50$

Use linear interpolation to estimate the median and the interquartile range.
Step-by-step solution
1. Step 1
  For $n = 50$ , the median is the value at position $n/2 = 25$ (Edexcel convention for grouped data uses $n/2$ , $n/4$ , $3n/4$ — not $(n+1)/2$ ).
2. Step 2
  Locate the class containing position $25$ : cumulative frequency reaches $24$ at $40$ and $38$ at $50$ , so the median lies in the class $40 < x \le 50$ . Linear interpolation:
  $Q_2 = 40 + \dfrac{25 - 24}{38 - 24} \times 10 = 40 + \dfrac{1}{14} \times 10 \approx 40.71$
3. Step 3
  $Q_1$ is at position $n/4 = 12.5$ — in the class $30 < x \le 40$ (which goes from CF $12$ to $24$ ).
  $Q_1 = 30 + \dfrac{12.5 - 12}{24 - 12} \times 10 = 30 + \dfrac{0.5}{12} \times 10 \approx 30.42$
4. Step 4
  $Q_3$ at position $3n/4 = 37.5$ — in the class $40 < x \le 50$ (CF $24$ to $38$ ).
  $Q_3 = 40 + \dfrac{37.5 - 24}{38 - 24} \times 10 = 40 + \dfrac{13.5}{14} \times 10 \approx 49.64$
5. Step 5
  Interquartile range:
  $\text{IQR} = Q_3 - Q_1 \approx 49.64 - 30.42 = 19.22$
Answer
Median $\approx 40.7$ , $Q_1 \approx 30.4$ , $Q_3 \approx 49.6$ , IQR $\approx 19.2$ (3 sf).
Examiner tip
M1 for using $n/2$ , $n/4$ , $3n/4$ to locate the quartiles. M1 for setting up linear interpolation. A1 for $Q_2$ . A1 for $Q_1$ . A1 for $Q_3$ . A1 for IQR. The Edexcel convention is to use $n/4$ for grouped/continuous data; using $(n+1)/4$ here would be marked wrong.
4Coding — find original mean and SD (5 marks, S1)
Extended• Adapted from WST01/01 June 2023 Q5• coding, mean, standard deviation
Question
Data values $x$ are coded by $y = \dfrac{x - 200}{5}$ . After coding, $\bar{y} = 4$ and $\sigma_y = 2$ . Find $\bar{x}$ and $\sigma_x$ .
Step-by-step solution
1. Step 1
  Rearrange the coding: $y = \dfrac{x - 200}{5}$ means $x = 5y + 200$ .
2. Step 2
  For a linear transformation $x = ay + b$ : $\bar{x} = a\bar{y} + b$ and $\sigma_x = |a|\sigma_y$ .
3. Step 3
  Apply to the mean:
  $\bar{x} = 5 \times 4 + 200 = 220$
4. Step 4
  Apply to the standard deviation:
  $\sigma_x = 5 \times 2 = 10$
Answer
$\bar{x} = 220$ , $\sigma_x = 10$ .
Examiner tip
M1 for inverting the coding to express $x$ in terms of $y$ . M1 for the linear transformation rule for the mean. A1 for $\bar{x}$ . M1 for the SD rule (SD scales by $|a|$ , no shift). A1 for $\sigma_x$ . A frequent error: adding $200$ to $\sigma$ — but constants do not affect spread.
5Compare two data sets — choose statistics (5 marks, S1)
Extended• Adapted from WST01/01 June 2024 Q5• comparison, mean, SD, outliers
Question
Class A: $\bar{x}_A = 64$ , $\sigma_A = 8$ , all $20$ marks between $48$ and $80$ . Class B: median $= 62$ , $Q_1 = 50$ , $Q_3 = 72$ , with two extreme outliers at $15$ and $95$ . Compare the two classes in terms of (a) typical performance and (b) variability, giving reasons for your choice of statistics.
Step-by-step solution
1. Step 1
  Class A has no outliers — its mean and SD are reliable summaries. Class B has two extreme outliers, which would distort the mean and SD, so use the median and IQR instead.
2. Step 2
  Typical performance:
  $\bar{x}_A = 64, \quad \text{median}_B = 62$
3. Step 3
  Class A's typical mark ( $64$ ) is slightly higher than Class B's typical mark ( $62$ ). Both classes are similar on a 'centre of distribution' basis.
4. Step 4
  Variability:
  $\sigma_A = 8, \quad \text{IQR}_B = 72 - 50 = 22$
5. Step 5
  Class B has substantially greater spread (IQR $= 22$ ) than Class A ( $\sigma_A = 8$ , roughly equivalent to IQR $\approx 11$ for a normal). Even allowing for the IQR vs SD differences, Class B's middle $50\%$ spans twice the range of Class A's typical $\pm 1$ SD interval.
Answer
Class A's mean ( $64$ ) is slightly higher than Class B's median ( $62$ ). Class B has much greater spread (IQR $= 22$ ) than Class A ( $\sigma = 8$ ). Median/IQR are used for B because outliers would distort the mean/SD.
Examiner tip
B1 for noting that outliers in B make mean/SD unreliable. B1 for choosing median/IQR for B and mean/SD for A. B1 for the centre comparison. B1 for the spread comparison. B1 for context (in marks). Examiners reward statements like 'Class A is more consistent than Class B' over generic 'Class B has higher SD'.

$t$ (mins)	$0 \le t < 10$	$10 \le t < 20$	$20 \le t < 30$	$30 \le t < 40$
frequency $f$	$4$	$12$	$16$	$8$

Mark $\le$	$20$	$30$	$40$	$50$	$60$	$70$	$80$
Cumulative frequency	$4$	$12$	$24$	$38$	$44$	$48$	$50$

Key Formulae — Measures of location and spread

The formulae you need to memorise for measures of location and spread on the Pearson Edexcel IAL XMA01 / YMA01 paper, with every variable defined in plain English and a note on when to use it.

Mean from a frequency table
$\bar{x} = \dfrac{\sum fx}{\sum f}$
$x$
value (or class midpoint for grouped data)
$f$
frequency of that value/class
$\bar{x}$
sample mean
When to use
Mean of any frequency / grouped frequency data. For raw data this reduces to $\bar{x} = \sum x / n$ . IS in the IAL formula booklet.
Example
With $\sum fx = 880$ and $\sum f = 40$ : $\bar{x} = 22$ .
Variance (shortcut)
$\sigma^{2} = \dfrac{\sum x^{2}}{n} - \bar{x}^{2}, \quad \sigma^{2} = \dfrac{\sum fx^{2}}{\sum f} - \bar{x}^{2}$
$\sigma^{2}$
(population) variance
$\sum x^{2}$
sum of squared values
$\bar{x}$
sample mean
When to use
Compute $\sigma^{2}$ from summary statistics. Then take square root for $\sigma$ . IS in the IAL formula booklet under S1. Avoid the deviation-from-mean form unless the question gives you that directly.
Example
$\sum x^{2} = 36{,}450$ , $n = 20$ , $\bar{x} = 42.5$ : $\sigma^{2} = 1822.5 - 1806.25 = 16.25$ , so $\sigma \approx 4.03$ .
Linear coding of mean and SD
$y = ax + b \Rightarrow \bar{y} = a\bar{x} + b, \;\;\sigma_y = |a|\sigma_x$
$a$
scale factor
$b$
shift constant
When to use
When data has been coded for ease of calculation or to recover the original mean / SD from a coded summary. NOT in the formula booklet — memorise. The key fact: shifts ( $b$ ) do not change spread.
Example
If $y = (x - 200)/5$ then $x = 5y + 200$ : $\bar{x} = 5\bar{y} + 200$ and $\sigma_x = 5\sigma_y$ .
Interquartile range
$\text{IQR} = Q_3 - Q_1$
$Q_1$
lower quartile (25th percentile)
$Q_3$
upper quartile (75th percentile)
When to use
Robust measure of spread — preferred to SD when data contains outliers. NOT in the formula booklet but standard knowledge.
Example
$Q_1 = 30.4$ , $Q_3 = 49.6$ : IQR $= 19.2$ .
Pearson's coefficient of skewness
$\text{Skew} = \dfrac{3(\bar{x} - \text{median})}{\sigma}$
$\bar{x}$
mean
$\text{median}$
median
$\sigma$
standard deviation
When to use
Numerical test of symmetry. Positive $\Rightarrow$ positive skew, negative $\Rightarrow$ negative skew. IS in the IAL formula booklet.
Example
$\bar{x} = 22$ , median $= 24$ , $\sigma = 5$ : skew $= 3(22 - 24)/5 = -1.2$ (negative skew).

Key Definitions and Keywords — Measures of location and spread

Definitions to memorise and the exact keywords mark schemes credit for measures of location and spread answers — sharpened from recent examiner reports for the 2026 Pearson Edexcel IAL XMA01 / YMA01 sitting.

Mean
Examiner keyword
Arithmetic average $\bar{x} = \sum x / n$ (or $\sum fx / \sum f$ ). Most efficient single-number summary of centre when data is symmetric and has no outliers; sensitive to extreme values.
Median
Examiner keyword
The middle value when data is ordered. For raw data of size $n$ : median is the $((n+1)/2)$ -th value if $n$ odd, average of $(n/2)$ -th and $((n/2)+1)$ -th values if $n$ even. For grouped/continuous data, Edexcel uses the $(n/2)$ -th by linear interpolation. Robust to outliers.
Mode
The most frequent value (or class for grouped data). A data set may be unimodal, bimodal or have no mode. Often the least informative of the three measures of centre but useful for categorical data.
Quartiles
Examiner keyword
Three values that split ordered data into four equal parts. $Q_1$ = lower quartile (25th percentile), $Q_2$ = median (50th), $Q_3$ = upper quartile (75th). For grouped data, located at $n/4$ , $n/2$ , $3n/4$ by linear interpolation.
Interquartile range (IQR)
Examiner keyword
$Q_3 - Q_1$ . Measures spread of the middle $50\%$ of the data. Robust to outliers — preferred over SD when outliers are present.
Variance and standard deviation
Examiner keyword
Variance $\sigma^{2}$ is the average of squared deviations from the mean. SD $\sigma = \sqrt{\sigma^{2}}$ has the same units as $x$ . S1 uses the population formula (divide by $n$ , not $n - 1$ ).
Coding
Examiner keyword
Replacing each data value $x$ by $y = (x - a)/h$ (or any linear transformation) to make arithmetic easier. The mean shifts and scales the same way; the SD scales but does not shift.

Common Mistakes and Misconceptions — Measures of location and spread

The traps other students keep falling into on measures of location and spread questions — taken from recent Pearson Edexcel IAL XMA01 / YMA01 examiner reports and mark schemes — and how to avoid them.

✕Using class boundaries instead of midpoints in $\sum fx$
WST01/01 examiner reports — annual
Why it happens
Students copy the boundary from the table without computing the midpoint.
How to avoid it
For each class $[a, b)$ , the midpoint is $(a + b)/2$ . Write the midpoint as a separate column before computing $\sum fx$ .
✕Dividing by $n - 1$ for S1 standard deviation
Why it happens
Calculator default setting or memory of statistics from another course.
How to avoid it
S1 uses the population formula $\sigma^{2} = \sum x^{2}/n - \bar{x}^{2}$ . Check your calculator's $\sigma_n$ vs $\sigma_{n-1}$ button. The formula booklet uses $n$ .
✕Claiming the SD changes when a constant is added
WST01/01 examiner reports — recurring
Why it happens
Confusion between shift and scale.
How to avoid it
If $y = x + b$ : SD is unchanged. If $y = ax$ : SD multiplies by $|a|$ . If $y = ax + b$ : SD multiplies by $|a|$ only — the $b$ has no effect.
✕Using $(n + 1)/4$ for $Q_1$ on grouped data
Why it happens
The $(n+1)/4$ rule is correct for raw discrete data but not for the Edexcel-S1 grouped-data convention.
How to avoid it
For grouped/continuous data, Edexcel uses $n/4$ , $n/2$ , $3n/4$ with linear interpolation. For raw data, use $(n+1)/4$ . The question wording usually makes the data type clear.
✕Omitting units from the final mean or SD
Why it happens
Students focus on the number.
How to avoid it
Always quote the unit (kg, min, marks, m). A-marks for the final answer often depend on units being present.
✕Saying 'A has bigger SD than B' without context
Why it happens
Generic statistical phrasing.
How to avoid it
Always finish a comparison with what it means in context: 'Class A is more consistent in mark — typical deviation $8$ marks vs $22$ in Class B.' Examiners look for the named variable in the sentence.

Measures of location and spread — frequently asked questions

The things students keep getting wrong in this sub-topic, answered.

Measures of location and spread

Short Notes - Measures of location and spread

Detailed Study Notes

At a glance

What you’ll learn

Quick recap

Memorise this

How it’s examined

Take this whole topic with you

1Mean from a grouped frequency table (4 marks, S1)

2Variance and SD using the shortcut formula (5 marks, S1)

3Quartiles by linear interpolation (6 marks, S1)

4Coding — find original mean and SD (5 marks, S1)

5Compare two data sets — choose statistics (5 marks, S1)

Mean from a frequency table

Variance (shortcut)

Linear coding of mean and SD

Interquartile range

Pearson's coefficient of skewness

Mean

Median

Mode

Quartiles

Interquartile range (IQR)

Variance and standard deviation

Coding

✕Using class boundaries instead of midpoints in ∑fx\sum fx∑fx

✕Dividing by n−1n - 1n−1 for S1 standard deviation

✕Claiming the SD changes when a constant is added

✕Using (n+1)/4(n + 1)/4(n+1)/4 for Q1Q_1Q1​ on grouped data

✕Omitting units from the final mean or SD

✕Saying 'A has bigger SD than B' without context

Measures of location and spread — frequently asked questions

✕Using class boundaries instead of midpoints in $\sum fx$

✕Dividing by $n - 1$ for S1 standard deviation

✕Using $(n + 1)/4$ for $Q_1$ on grouped data