Probability Distributions Comparisons

Statistics AQA
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ST20: Probability Distributions & Comparisons

Edexcel 1ST0 & AQA 8382

Explores key probability distributions including the binomial and normal distributions, the 68–95–99.7 rule for normally distributed data, and techniques for comparing data sets using summary statistics and standardised scores.

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Probability Distributions & Comparisons

Explores key probability distributions including the binomial and normal distributions, the 68–95–99.7 rule for normally distributed data, and techniques for comparing data sets using summary statistics and standardised scores.

Key Fact: A probability distribution lists all possible outcomes with their associated probabilities; the total probability always sums to 1
Key Fact: Binomial distribution models the number of successes in n independent trials, each with the same probability p of success; P(X = r) = ²ⁿCᵣ × pʳ × (1−p)ⁿʻʳ where X~B(n, p)
Key Fact: For a binomial distribution: mean = np and variance = np(1−p); these are the expected value and spread
Key Fact: The normal distribution is a continuous bell-shaped curve, fully defined by its mean (μ) and standard deviation (σ); notation X~N(μ, σ²)
Key Fact: The 68–95–99.7 rule: approximately 68% of data lies within 1σ of the mean, 95% within 2σ, and 99.7% within 3σ
Key Fact: The normal distribution is symmetric about the mean; the mean, median and mode are all equal
Key Fact: Standardised score (z-score) = (value − mean) ÷ standard deviation; it measures how many standard deviations a value is from the mean
Key Fact: Standardising allows comparison between different distributions by converting values to a common N(0,1) scale
Key Fact: When comparing two data sets, compare medians or means for location, and IQR or standard deviation for spread
Key Fact: Use the context when comparing: always state which group is higher/lower and by how much, using summary statistics with units
Key Fact: A binomial situation requires: fixed number of trials, two outcomes (success/failure), independent trials, and constant probability of success
Key Fact: For normal distribution calculations, use z-tables or a calculator to find areas under the curve between given z-values

📋 Key Vocabulary and Concepts

For Probability Distributions & Comparisons, you must know:

❓ Practice Questions

Q: A fair coin is tossed 10 times. What is the probability of getting exactly 7 heads?

Q: Heights of a population are normally distributed with mean 170 cm and standard deviation 6 cm. Use the 68–95–99.7 rule to estimate the percentage of people between 158 cm and 182 cm.

Q: Student A scores 72 on a test with mean 60 and SD 8. Student B scores 78 on a test with mean 70 and SD 5. Who performed better relative to their class?

Q: A dice is rolled 20 times. Using a binomial model with p = 1/6 for rolling a 6, find the expected number of sixes and the variance.

Q: Explain why a binomial distribution would not be appropriate for modelling the number of goals scored in a football match.

✅ Answers

  1. X~B(10, 0.5). P(X=7) = ¹⁰C₇ × 0.5⁷ × 0.5³ = 120 × 0.0078125 × 0.125 = 0.1171875 ≈ 0.117
  2. 158 = 170 − 12 = mean − 2σ. 182 = 170 + 12 = mean + 2σ. By the 68–95–99.7 rule, approximately 95% of data lies within 2σ of the mean.
  3. z_A = (72−60)/8 = 1.5. z_B = (78−70)/5 = 1.6. Student B performed better as their z-score is higher.
  4. Mean = np = 20 × 1/6 ≈ 3.33. Variance = np(1−p) = 20 × (1/6) × (5/6) ≈ 2.78.
  5. The binomial requires a fixed number of independent trials with constant probability. Goals in a football match do not have a fixed number of trials, the probability of scoring likely changes during the match, and goals may not be independent events (momentum effects).

🎯 Exam Tips

📝 Exam Technique

GCSE Statistics Exam Tips — Probability Distributions & Comparisons:
1. For Probability Distributions & Comparisons questions, show every step of your working clearly — method marks count even if the final answer is wrong
2. Check your answer makes sense in context (estimation, units, reasonableness)
3. Use correct mathematical notation and state formulae before substituting values
4. If a Probability Distributions & Comparisons question asks you to 'prove' or 'show', write a logical chain of reasoning with a conclusion line
5. For problem-solving, identify the topic first, then recall the relevant method

⚠️ Common Errors

✗ Using σ² instead of σ when applying the 68–95–99.7 rule ✓ The rule uses standard deviation σ, not variance σ²; mean ± 2σ covers 95%

✗ Confusing discrete and continuous distributions ✓ Binomial is discrete (count successes); normal is continuous (measure values on a scale)

✗ Comparing raw scores across different distributions without standardising ✓ Convert to z-scores first to make fair comparisons on a common scale

✗ Assuming all data is normally distributed without checking ✓ Only use the normal distribution when told the data is normally distributed or when the distribution is symmetric and bell-shaped

✍️ Model Answer

Full-Mark Response

The masses of apples from Farm A are normally distributed with mean 150 g and standard deviation 12 g. The masses of apples from Farm B are normally distributed with mean 160 g and standard deviation 20 g. An apple from Farm A has mass 174 g and an apple from Farm B has mass 180 g. Which apple is relatively heavier? Show your working.

Standardise each apple’s mass using z = (value − mean) ÷ SD. Farm A: z = (174 − 150) ÷ 12 = 24 ÷ 12 = 2.0 Farm B: z = (180 − 160) ÷ 20 = 20 ÷ 20 = 1.0 The Farm A apple is 2.0 standard deviations above its mean, while the Farm B apple is only 1.0 standard deviation above its mean. Therefore the Farm A apple is relatively heavier compared to its distribution.

📊 AO Deep Dive

Assessment Objective Analysis

AO1 (Knowledge & Understanding): Demonstrate knowledge and understanding of probability distributions & comparisons, including data collection, presentation and calculation techniques relevant to Edexcel 1ST0 & AQA 8382.

AO2 (Application): Apply knowledge and understanding of probability distributions & comparisons to interpret data, reason statistically and draw conclusions in context.

AO3 (Evaluation): Evaluate statistical methods and conclusions, assessing appropriateness, reliability, validity and bias through the statistical enquiry cycle.

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