What are approximations in the context of Statistics 2 for Edexcel International A Levels?

In Statistics 2 for Edexcel International A Levels, approximations refer to methods used to estimate values or probabilities when exact calculations are complex or impossible. This often involves using simpler models or distributions to approximate more complex ones, such as using the normal distribution to approximate binomial or Poisson distributions under certain conditions.

When can the normal distribution be used as an approximation for the binomial distribution?

The normal distribution can be used to approximate the binomial distribution when the number of trials (n) is large and the probability of success (p) is neither very close to 0 nor 1. A common rule of thumb is that both np and n(1-p) should be greater than 5. This allows for the binomial distribution to be approximated by a normal distribution with mean np and variance np(1-p).

How do you apply continuity correction when using normal approximation for discrete distributions?

When using the normal distribution to approximate a discrete distribution like the binomial or Poisson, a continuity correction is applied to account for the fact that the normal distribution is continuous. This involves adjusting the discrete x-value by 0.5 in the direction that makes the approximation more accurate. For example, to approximate P(X ≤ x) using a normal distribution, you would calculate P(X ≤ x + 0.5).

What is the role of the Poisson distribution in approximations within Statistics 2?

In Statistics 2, the Poisson distribution is often used as an approximation for the binomial distribution when the number of trials is large, and the probability of success is small. This is particularly useful when n is large, and p is small such that np is moderate. The Poisson approximation simplifies calculations by using the parameter λ = np.

Why are approximations important in statistical analysis for Edexcel International A Levels?

Approximations are crucial in statistical analysis because they simplify complex problems, making them more manageable and easier to solve. In the context of Edexcel International A Levels, they allow students to apply mathematical concepts to real-world situations where exact calculations are impractical, thus enhancing their problem-solving skills and understanding of statistical methods.

Why is "Omitting the continuity correction" a common mistake in Approximations?

Why it happens: Students forget the 0.5 when going discrete continuous. How to avoid it: Write the original probability, then explicitly write the continuity-corrected version, e.g. 'P(X 35) P(Y < 35.5)' on its own line. Source: WST02/01 examiner reports — annual top error

Why is "Continuity correction in the WRONG direction" a common mistake in Approximations?

Why it happens: Confusion about whether to add or subtract 0.5. How to avoid it: Rule: or rounds OUTWARD. P(X 35) P(Y 9.5) (outward = downward). For strict inequalities, first rewrite as the equivalent / .

Why is "Using the wrong approximation" a common mistake in Approximations?

Why it happens: Not checking the conditions. How to avoid it: Always check the conditions before approximating. B(500, 0.004): p small Poisson. B(80, 0.5): np large Normal. Both? Either works but mark scheme usually accepts either.

Why is "Using $\sigma^{2} = np$ (Poisson) when approximating binomial by normal" a common mistake in Approximations?

Why it happens: Mixing the two approximations. How to avoid it: Binomial Normal: variance = np(1-p). Poisson Normal: variance = . Different formulae — keep them separate.

Why is "Treating $P(X = k)$ as $P(Y = k)$ in normal approximation" a common mistake in Approximations?

Why it happens: P(Y = k) = 0 for any continuous Y, so this gives an absurd zero answer. How to avoid it: P(X = k) P(k - 0.5 < Y < k + 0.5) — a non-zero interval probability. Source: WST02/01 examiner reports

Statistics 2Edexcel International A Levels Mathematics (XMA01-YMA01)

Approximations

Q: Why is "Continuity correction in the WRONG direction" a common mistake in Approximations?

Why it happens: Confusion about whether to add or subtract 0.5. How to avoid it: Rule: or rounds OUTWARD. P(X 35) P(Y 9.5) (outward = downward). For strict inequalities, first rewrite as the equivalent / .

Q: Why is "Using the wrong approximation" a common mistake in Approximations?

Why it happens: Not checking the conditions. How to avoid it: Always check the conditions before approximating. B(500, 0.004): p small Poisson. B(80, 0.5): np large Normal. Both? Either works but mark scheme usually accepts either.

Q: Why is "Using $\sigma^{2} = np$ (Poisson) when approximating binomial by normal" a common mistake in Approximations?

Why it happens: Mixing the two approximations. How to avoid it: Binomial Normal: variance = np(1-p). Poisson Normal: variance = . Different formulae — keep them separate.

Q: Why is "Treating $P(X = k)$ as $P(Y = k)$ in normal approximation" a common mistake in Approximations?

Why it happens: P(Y = k) = 0 for any continuous Y, so this gives an absurd zero answer. How to avoid it: P(X = k) P(k - 0.5 < Y < k + 0.5) — a non-zero interval probability. Source: WST02/01 examiner reports

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Approximations Study Notes — Edexcel IAL Mathematics S2 (WST02/01, 2018 spec — 2026 onwards)

Three approximations: $B(n,p) \to \text{Po}(np)$ (rare events), $B(n,p) \to N(np, np(1-p))$ (large $n$ , mid $p$ ), $\text{Po}(\lambda) \to N(\lambda, \lambda)$ (large $\lambda$ ). The continuity correction $\pm 0.5$ is mandatory for any discrete-to-continuous step. Conditions, parameters, and the correction rules form a coherent toolkit.

What you’ll learn

Mapped to the Cambridge IGCSE XMA01-YMA01 syllabus (2026 onwards).

S2 3.1 — State the conditions for each of the three approximations.
S2 3.2 — Choose the appropriate approximation for given $(n, p)$ or $\lambda$ .
S2 3.3 — Calculate the parameters of the approximating distribution.
S2 3.4 — Apply the continuity correction when going from discrete to continuous.
S2 3.5 — Justify the choice of approximation in context.

Why approximate?

Binomial calculations for large $n$ are awkward; normal calculations use familiar tables.

Computational motivation. Computing $\binom{500}{12}$ on a calculator is fine, but historically (and on tables-only papers) the binomial PMF becomes impractical for large $n$ . Approximations replace the awkward distribution with a simpler one that:

Is defined in the formula booklet (normal tables, Poisson tables).
Has familiar mean and variance.
Is close in shape to the original.

Modern context. Calculators can evaluate binomial and Poisson directly, but Edexcel still tests approximations because:

They show conceptual understanding of distribution shape.
Hypothesis tests (later topic) often use normal approximations for tractability.
The continuity correction is a hallmark of careful probabilistic thinking.

Hierarchy.

Binomial $\to$ Poisson when $p$ is small ( $\lambda$ stays moderate).
Binomial $\to$ Normal when $p$ is mid-range (mean large).
Poisson $\to$ Normal when $\lambda$ is large.
If $B(n, p)$ has both $p$ small AND $np$ large, BOTH Poisson and Normal apply — and indeed $\text{Po}(\lambda)$ then approximates $N(\lambda, \lambda)$ .

Approximate when exact is awkward.
Approximate when only tables are available.
Modern reason: hypothesis tests use approximations.

Binomial $\to$ Poisson

$B(n, p) \approx \text{Po}(np)$ for $n \ge 50$ and $p \le 0.1$ .

Conditions. $n$ large, $p$ small. Edexcel rule of thumb: $n \ge 50$ AND $p \le 0.1$ .

Parameter. $\lambda = np$ .

Why it works. For small $p$ , $n p(1-p) \approx np$ , so the binomial's variance is approximately equal to its mean — matching the Poisson identity. The binomial PMF approaches the Poisson PMF in the limit $n \to \infty, p \to 0, np = \lambda$ fixed.

Worked example. $X \sim B(200, 0.015)$ .

Conditions: $n = 200 \ge 50$ , $p = 0.015 \le 0.1$ . Both satisfied.
Approximation: $X \approx \text{Po}(3)$ (since $np = 3$ ).
$P(X = 3) \approx e^{-3} \cdot 27/6 \approx 0.2240$ .

No continuity correction. Both distributions are discrete — the correction is only for discrete $\to$ continuous.

Accuracy. Excellent for the conditions above. Error decreases as $p \to 0$ .

Conditions: $n \ge 50$ , $p \le 0.1$ .
Approx: $\text{Po}(np)$ .
NO continuity correction (both discrete).

Binomial $\to$ Normal

$B(n, p) \approx N(np, np(1-p))$ for $np > 5$ and $n(1-p) > 5$ .

Conditions. $np > 5$ AND $n(1-p) > 5$ . Some textbooks: $\ge 5$ — Edexcel mark schemes are usually generous.

Parameters.

Mean $\mu = np$ .
Variance $\sigma^{2} = np(1-p)$ .
SD $\sigma = \sqrt{np(1-p)}$ .

Why it works. By the Central Limit Theorem, a sum of $n$ independent identically-distributed Bernoullis approaches a normal distribution as $n \to \infty$ . The conditions ensure both tails are reasonably populated.

Worked example. $X \sim B(80, 0.4)$ , find $P(X \le 35)$ .

Conditions: $np = 32 > 5$ , $n(1-p) = 48 > 5$ . Use normal.
$\mu = 32$ , $\sigma^{2} = 19.2$ , $\sigma \approx 4.382$ .
Continuity: $P(X \le 35) \approx P(Y \le 35.5)$ .
Standardise: $z = (35.5 - 32)/4.382 \approx 0.80$ .
$\Phi(0.80) \approx 0.7881$ .

Continuity correction. Mandatory. $P(X \le 35) \to P(Y \le 35.5)$ .

Why $35.5$ ? The discrete bar at $X = 35$ spans the interval $34.5$ to $35.5$ on the continuous axis. Including the whole bar means going up to $35.5$ .

Conditions: $np > 5$ AND $n(1-p) > 5$ .
Approx: $N(np, np(1-p))$ .
Continuity correction required.

Poisson $\to$ Normal

$\text{Po}(\lambda) \approx N(\lambda, \lambda)$ for $\lambda > 10$ .

Condition. $\lambda > 10$ . (Some texts $\ge 10$ or $> 15$ — Edexcel mark schemes accept either.)

Parameters. Mean = variance = $\lambda$ . So $\mu = \lambda$ , $\sigma^{2} = \lambda$ , $\sigma = \sqrt{\lambda}$ .

Why it works. The Poisson is a limit of binomials; for large $\lambda$ , the Poisson PMF becomes approximately bell-shaped. The skewness ( $1/\sqrt{\lambda}$ ) shrinks to zero, and the symmetry of the normal becomes a good fit.

Worked example. $X \sim \text{Po}(40)$ , find $P(X > 45)$ .

Condition: $\lambda = 40 > 10$ . Use normal.
$\mu = 40$ , $\sigma^{2} = 40$ , $\sigma \approx 6.325$ .
Continuity: $X > 45 \Rightarrow X \ge 46 \to Y > 45.5$ .
Standardise: $z = (45.5 - 40)/6.325 \approx 0.87$ .
$P(X > 45) \approx 1 - \Phi(0.87) \approx 0.192$ .

Variance trap. Variance is $\lambda$ , not $\lambda^{2}$ or anything else. SD $= \sqrt{\lambda}$ .

Condition: $\lambda > 10$ .
Approx: $N(\lambda, \lambda)$ .
$\sigma = \sqrt{\lambda}$ .
Continuity correction required.

The continuity correction in detail

$\pm 0.5$ outward whenever discrete is approximated by continuous.

Why needed. A discrete count $X$ takes integer values; the binomial / Poisson PMF assigns positive probability to each integer. The normal density is continuous; $P(Y = k) = 0$ for any single $k$ . So we replace each integer $k$ with the interval $(k - 0.5, k + 0.5)$ .

Rules.

Discrete	Continuous
$P(X = k)$	$P(k - 0.5 < Y < k + 0.5)$
$P(X \le k)$	$P(Y < k + 0.5)$
$P(X \ge k)$	$P(Y > k - 0.5)$
$P(X < k)$	$P(Y < k - 0.5)$ (since $X < k$ means $X \le k - 1$ )
$P(X > k)$	$P(Y > k + 0.5)$ (since $X > k$ means $X \ge k + 1$ )
$P(a \le X \le b)$	$P(a - 0.5 < Y < b + 0.5)$

Rule of thumb (outward). $\le$ and $\ge$ extend OUTWARD by $0.5$ . Strict $<$ and $>$ first rewrite as the equivalent non-strict on the integer side, then extend outward.

Why outward? To include the whole 'bar' at the boundary integer. For $X \le 35$ , the bar at $X = 35$ extends from $34.5$ to $35.5$ — so 'up to $35.5$ '.

Picture. Sketch the discrete bars vs the normal curve, shade the appropriate region, and you can never get the direction wrong.

Each integer $k$ $\to$ interval $(k - 0.5, k + 0.5)$ .
$\le$ and $\ge$ extend OUTWARD by $0.5$ .
Strict $<, >$ : rewrite as non-strict first.
Sketch to confirm direction.

Choosing the right approximation

Check conditions in order: Poisson first if $p$ small; otherwise Normal.

Decision tree.

Is the original distribution Poisson, large $\lambda$ ? $\lambda > 10 \Rightarrow$ Normal approximation.
Is the original binomial? Check:
- Small $p$ AND large $n$ : $p \le 0.1$ , $n \ge 50$ $\Rightarrow$ Poisson with $\lambda = np$ .
- Mid $p$ AND large $n$ : $np > 5$ , $n(1-p) > 5$ $\Rightarrow$ Normal with $\mu = np$ , $\sigma^{2} = np(1-p)$ .
- Both apply (e.g. $n = 500$ , $p = 0.04$ ): Poisson is usually preferred because $\lambda = 20 > 10$ , so Po-to-N can be chained.
Neither applies? Use the exact distribution.

Why prefer Poisson when $p$ is small?

Better fit to the skewed shape (normal is symmetric).
One parameter instead of two — simpler.
Cumulative Poisson tables in the booklet handle the lookup.

When normal is essential. $B(n, p)$ with $p$ near $0.5$ — Poisson would be a terrible fit (Poisson has mean = variance, but $B(n, 0.5)$ has variance = $n/4$ , which is far from $n/2 = \lambda$ ).

Small $p$ $\to$ Poisson.
Mid $p$ , large $n$ $\to$ Normal.
Large $\lambda$ Poisson $\to$ Normal.
Justify in working — examiners look for condition checks.

How it’s examined

Approximations appear every WST02/01 session — typically a $5$ - $8$ mark question requiring (1) condition check, (2) parameter calculation, (3) continuity correction, (4) standardisation, (5) probability lookup. Mark schemes break the marks down across these steps; missing the continuity correction usually costs $1$ - $2$ marks. Multi-part questions often ask 'state the approximation' for part (a), then apply it for part (b). The justification step ('state conditions') routinely earns a B-mark on its own.

Sources: Pearson Edexcel International Advanced Level Mathematics specification (2018 — current for 2026 onwards); Edexcel IAL Mathematical Formulae and Statistical Tables (2018); WST02/01 Examiner Reports — January 2023, June 2023, January 2024, June 2024. Last reviewed 2026-05-12.

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Step-by-step worked examples — Approximations

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

1Binomial $\to$ Poisson approximation (5 marks, S2)
Core• Adapted from WST02/01 January 2024 Q4• approximation, binomial, poisson
Question
A factory produces components with a $1.5\%$ defect rate. A sample of $200$ components is inspected. Using a suitable approximation, find the probability that the sample contains exactly $3$ defectives.
Step-by-step solution
1. Step 1
  Exact: $X \sim B(200, 0.015)$ . Check approximation conditions: $n = 200 \ge 50$ (large), $p = 0.015 \le 0.1$ (small). Use $\text{Po}$ .
2. Step 2
  Parameter: $\lambda = np = 200 \times 0.015 = 3$ .
  $X \approx \text{Po}(3)$
3. Step 3
  Apply PMF:
  $P(X = 3) \approx e^{-3} \dfrac{3^{3}}{3!} = e^{-3} \dfrac{27}{6}$
4. Step 4
  $e^{-3} \approx 0.04979$ , so $P(X = 3) \approx 0.04979 \times 4.5 \approx 0.2240$ .
Answer
$P(X = 3) \approx 0.2240$ ( $4$ dp).
Examiner tip
B1 for justifying the approximation (large $n$ , small $p$ ). B1 for $\lambda = 3$ . M1 for Poisson PMF, A1 for $0.2240$ . Mark schemes accept $0.224$ to $3$ sf. Note NO continuity correction needed — both are discrete.
2Binomial $\to$ Normal with continuity correction (6 marks, S2)
Extended• Adapted from WST02/01 June 2024 Q6• approximation, binomial, normal, continuity
Question
$X \sim B(80, 0.4)$ . Using a suitable approximation, find $P(X \le 35)$ .
Step-by-step solution
1. Step 1
  Check conditions: $np = 32 > 5$ , $n(1-p) = 48 > 5$ . Both large. Normal approximation valid.
2. Step 2
  Parameters:
  $\mu = np = 32, \quad \sigma^{2} = np(1-p) = 80 \times 0.4 \times 0.6 = 19.2$
3. Step 3
  $\sigma = \sqrt{19.2} \approx 4.382$ . So $X \approx N(32, 19.2)$ .
4. Step 4
  Continuity correction (discrete to continuous):
  $P(X \le 35) \approx P(Y \le 35.5)$
5. Step 5
  Standardise:
  $z = \dfrac{35.5 - 32}{4.382} = \dfrac{3.5}{4.382} \approx 0.7988$
6. Step 6
  From normal tables: $\Phi(0.80) \approx 0.7881$ . (Or $\Phi(0.7988) \approx 0.7878$ .)
Answer
$P(X \le 35) \approx 0.788$ ( $3$ sf).
Examiner tip
B1 for checking conditions. B1 for $\mu = 32$ , $\sigma^{2} = 19.2$ . M1 for continuity correction $\le 35.5$ . M1 for standardisation. A1 for $z \approx 0.80$ . A1 for $0.788$ . The $+0.5$ continuity correction is a MARK-EARNING step — omitting it is the most common error.
3Poisson $\to$ Normal with continuity correction (6 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q6• approximation, poisson, normal, continuity
Question
$X \sim \text{Po}(40)$ . Using a suitable approximation, find $P(X > 45)$ .
Step-by-step solution
1. Step 1
  Check: $\lambda = 40 > 10$ . Normal approximation valid.
2. Step 2
  Parameters: $\mu = \sigma^{2} = \lambda = 40$ , so $\sigma = \sqrt{40} \approx 6.325$ .
  $X \approx N(40, 40)$
3. Step 3
  Continuity: $P(X > 45) = P(X \ge 46) \approx P(Y > 45.5)$ .
4. Step 4
  Standardise:
  $z = \dfrac{45.5 - 40}{6.325} = \dfrac{5.5}{6.325} \approx 0.8696$
5. Step 5
  Probability:
  $P(X > 45) \approx 1 - \Phi(0.87) \approx 1 - 0.8078 = 0.1922$
Answer
$P(X > 45) \approx 0.192$ ( $3$ sf).
Examiner tip
B1 for $\lambda > 10$ check. B1 for $\mu = \sigma^{2} = 40$ . M1 for continuity correction (note $X > 45 \Rightarrow Y > 45.5$ ). M1 for standardisation, A1 for $z \approx 0.87$ . A1 for $0.192$ . Common error: writing $P(X > 45) \approx P(Y > 44.5)$ — wrong direction.
4Continuity corrections for $<, \le, >, \ge, =$ (5 marks, S2)
Extended• Adapted from WST02/01 June 2024 Q7• approximation, continuity correction
Question
$X \sim B(100, 0.5) \approx N(50, 25)$ . Using the normal approximation with continuity correction, write down the equivalent normal probability for each: (a) $P(X = 50)$ , (b) $P(X < 45)$ , (c) $P(X \le 45)$ , (d) $P(X > 55)$ , (e) $P(45 \le X \le 55)$ .
Step-by-step solution
1. Step 1
  $\sigma = 5$ .
2. Step 2
  (a) Single point $X = 50$ uses an interval of width $1$ centred on $50$ :
  $P(X = 50) \approx P(49.5 < Y < 50.5)$
3. Step 3
  (b) $X < 45$ means $X \le 44$ (discrete):
  $P(X < 45) = P(X \le 44) \approx P(Y < 44.5)$
4. Step 4
  (c) $X \le 45$ :
  $P(X \le 45) \approx P(Y < 45.5)$
5. Step 5
  (d) $X > 55$ means $X \ge 56$ :
  $P(X > 55) = P(X \ge 56) \approx P(Y > 55.5)$
6. Step 6
  (e) $45 \le X \le 55$ uses BOTH endpoints widened by $0.5$ :
  $P(45 \le X \le 55) \approx P(44.5 < Y < 55.5)$
Answer
(a) $P(49.5 < Y < 50.5)$ . (b) $P(Y < 44.5)$ . (c) $P(Y < 45.5)$ . (d) $P(Y > 55.5)$ . (e) $P(44.5 < Y < 55.5)$ .
Examiner tip
B1 for each correct correction. The trick: any ' $\le$ ' or ' $\ge$ ' rounds OUTWARD ( $\pm 0.5$ ). Any ' $<$ ' or ' $>$ ' first rewrites as the equivalent $\le / \ge$ , then rounds outward. A single equality $X = k$ becomes $k - 0.5 < Y < k + 0.5$ (interval of width $1$ ).
5Justifying the choice of approximation (4 marks, S2)
Extended• Adapted from WST02/01 January 2024 Q9• approximation, justification, context
Question
A geneticist studies a rare gene mutation that occurs in $0.4\%$ of the population. A random sample of $500$ people is screened. (a) State the exact distribution of $X$ , the number of carriers. (b) Justify the use of a Poisson approximation. (c) Compare its accuracy with a normal approximation, briefly.
Step-by-step solution
1. Step 1
  (a) Independent screenings, fixed $n = 500$ , constant $p = 0.004$ .
  $X \sim B(500, 0.004)$
2. Step 2
  (b) Conditions for Poisson approx: $n = 500$ is large ( $\ge 50$ ), $p = 0.004$ is small ( $\le 0.1$ ). Both satisfied. Parameter $\lambda = np = 2$ .
3. Step 3
  (c) Normal approx requires $np > 5$ and $n(1-p) > 5$ . Here $np = 2 < 5$ , so the normal approximation FAILS. The Poisson approximation is appropriate; the normal is not.
4. Step 4
  Practical comment: with $\lambda = 2$ , the distribution is heavily right-skewed (small mean, long tail). The normal is symmetric and would badly underestimate upper-tail probabilities. The Poisson handles skewness correctly.
Answer
(a) $X \sim B(500, 0.004)$ . (b) $n$ large, $p$ small $\Rightarrow \text{Po}(\lambda = 2)$ . (c) Normal fails ( $np = 2 < 5$ ) and would be inaccurate due to skewness.
Examiner tip
(a) B1. (b) M1 for stating conditions, A1 for $\lambda = 2$ . (c) B1 for either ( $np < 5$ ) or skewness comment. Choice of approximation is a recurring exam theme — know all three condition sets cold.

Key Formulae — Approximations

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

Binomial $\to$ Poisson approximation
$B(n, p) \approx \text{Po}(np) \quad \text{when } n \text{ large, } p \text{ small}$
$n$
number of trials
$p$
success probability
When to use
Edexcel rule of thumb: $n \ge 50$ AND $p \le 0.1$ . NOT in the formula booklet — memorise the conditions.
Example
$B(200, 0.015) \approx \text{Po}(3)$ . $P(X = 3) \approx e^{-3} 3^{3}/3! \approx 0.2240$ .
Binomial $\to$ Normal approximation
$B(n, p) \approx N(np, np(1-p)) \quad \text{when } np > 5 \text{ and } n(1-p) > 5$
$n$
number of trials
$p$
success probability
When to use
Edexcel rule of thumb: $np > 5$ AND $n(1-p) > 5$ (or sometimes $\ge 5$ ). NOT in the booklet — memorise.
Example
$B(80, 0.4)$ : $np = 32, n(1-p) = 48$ , both $> 5$ . Use $N(32, 19.2)$ .
Poisson $\to$ Normal approximation
$\text{Po}(\lambda) \approx N(\lambda, \lambda) \quad \text{when } \lambda > 10$
$\lambda$
Poisson parameter
When to use
Edexcel rule of thumb: $\lambda > 10$ (some texts $\ge 10$ or $> 15$ ). NOT in the booklet — memorise.
Example
$\text{Po}(40) \approx N(40, 40)$ . $\sigma = \sqrt{40}$ .
Continuity correction
$P(X \le k) \approx P(Y \le k + 0.5), \;\; P(X \ge k) \approx P(Y \ge k - 0.5)$
$X$
discrete approximant (binomial or Poisson)
$Y$
continuous normal approximation
$k$
integer boundary
When to use
Whenever you approximate a discrete distribution by a continuous one. NOT in the booklet — memorise the rules.
Example
$P(X \le 35) \approx P(Y \le 35.5)$ . $P(X = 50) \approx P(49.5 < Y < 50.5)$ .

Key Definitions and Keywords — Approximations

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

Approximation
Examiner keyword
Replacing one distribution with another (simpler or easier to compute) that is close in shape. Used when exact calculations are tedious or tables unavailable.
Continuity correction
Examiner keyword
Adjustment of $\pm 0.5$ applied when approximating a DISCRETE distribution (binomial / Poisson) by a CONTINUOUS one (normal). Necessary because a discrete bar of width $1$ is replaced by a smooth density.
Binomial $\to$ Poisson conditions
Examiner keyword
$n$ large, $p$ small (Edexcel: $n \ge 50$ AND $p \le 0.1$ ).
Binomial $\to$ Normal conditions
Examiner keyword
$np > 5$ AND $n(1-p) > 5$ . (Some textbooks: $np \ge 5$ AND $nq \ge 5$ — equivalent.)
Poisson $\to$ Normal conditions
Examiner keyword
$\lambda > 10$ . (Some references use $\lambda \ge 10$ or $\lambda > 15$ — Edexcel mark scheme is generous.)

Common Mistakes and Misconceptions — Approximations

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

✕Omitting the continuity correction
WST02/01 examiner reports — annual top error
Why it happens
Students forget the $\pm 0.5$ when going discrete $\to$ continuous.
How to avoid it
Write the original probability, then explicitly write the continuity-corrected version, e.g. ' $P(X \le 35) \approx P(Y < 35.5)$ ' on its own line.
✕Continuity correction in the WRONG direction
Why it happens
Confusion about whether to add or subtract $0.5$ .
How to avoid it
Rule: $\le$ or $\ge$ rounds OUTWARD. $P(X \le 35) \to P(Y < 35.5)$ (outward = upward). $P(X \ge 10) \to P(Y > 9.5)$ (outward = downward). For strict inequalities, first rewrite as the equivalent $\le / \ge$ .
✕Using the wrong approximation
Why it happens
Not checking the conditions.
How to avoid it
Always check the conditions before approximating. $B(500, 0.004)$ : $p$ small $\Rightarrow$ Poisson. $B(80, 0.5)$ : $np$ large $\Rightarrow$ Normal. Both? Either works but mark scheme usually accepts either.
✕Using $\sigma^{2} = np$ (Poisson) when approximating binomial by normal
Why it happens
Mixing the two approximations.
How to avoid it
Binomial $\to$ Normal: variance $= np(1-p)$ . Poisson $\to$ Normal: variance $= \lambda$ . Different formulae — keep them separate.
✕Treating $P(X = k)$ as $P(Y = k)$ in normal approximation
WST02/01 examiner reports
Why it happens
$P(Y = k) = 0$ for any continuous $Y$ , so this gives an absurd zero answer.
How to avoid it
$P(X = k) \approx P(k - 0.5 < Y < k + 0.5)$ — a non-zero interval probability.

Practice questions

Exam-style questions with step-by-step worked solutions. Try one before checking the method.

Past paper style quiz

Get a report showing which sub-topics you've nailed and which ones still need work.

Approximations — frequently asked questions

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

Approximations

Short Study Notes in the form of Slides

Detailed Study Notes

What you’ll learn

How it’s examined

1Binomial →\to→ Poisson approximation (5 marks, S2)

2Binomial →\to→ Normal with continuity correction (6 marks, S2)

3Poisson →\to→ Normal with continuity correction (6 marks, S2)

4Continuity corrections for <,≤,>,≥,=<, \le, >, \ge, =<,≤,>,≥,= (5 marks, S2)

5Justifying the choice of approximation (4 marks, S2)

Binomial →\to→ Poisson approximation

Binomial →\to→ Normal approximation

Poisson →\to→ Normal approximation

Continuity correction

Approximation

Continuity correction

Binomial →\to→ Poisson conditions

Binomial →\to→ Normal conditions

Poisson →\to→ Normal conditions

✕Omitting the continuity correction

✕Continuity correction in the WRONG direction

✕Using the wrong approximation

✕Using σ2=np\sigma^{2} = npσ2=np (Poisson) when approximating binomial by normal

✕Treating P(X=k)P(X = k)P(X=k) as P(Y=k)P(Y = k)P(Y=k) in normal approximation

Practice questions

Past paper style quiz

Approximations — frequently asked questions

1Binomial $\to$ Poisson approximation (5 marks, S2)

2Binomial $\to$ Normal with continuity correction (6 marks, S2)

3Poisson $\to$ Normal with continuity correction (6 marks, S2)

4Continuity corrections for $<, \le, >, \ge, =$ (5 marks, S2)

Binomial $\to$ Poisson approximation

Binomial $\to$ Normal approximation

Poisson $\to$ Normal approximation

Binomial $\to$ Poisson conditions

Binomial $\to$ Normal conditions

Poisson $\to$ Normal conditions

✕Using $\sigma^{2} = np$ (Poisson) when approximating binomial by normal

✕Treating $P(X = k)$ as $P(Y = k)$ in normal approximation