S1 // COMPLETE LEARNING SYSTEM
Statistics 1
The aim of this page is not simply to memorise formulae. You should be able to recognise statistical situations, select appropriate methods, perform calculations accurately, interpret results in context, and explain conclusions using correct mathematical language.
TOPIC 01 // STATISTICAL SAMPLING
Populations, Samples and Sampling
A statistical investigation begins by identifying the population about which information is required. In many real investigations it is impractical, expensive or impossible to collect data from every member of the population. A sample is therefore selected.
Population
The population is the complete set of individuals or objects about which the investigation is concerned.
If a school wants to investigate the average number of hours its students spend studying each week, the population could be all students in that school.
Sample
A sample is a subset of the population selected for observation. Statistical conclusions about the population are made using information obtained from the sample.
Census
A census collects information from every member of the population. Although a census eliminates sampling variation, it may be expensive, time-consuming or difficult to organise.
Sampling Frame
A sampling frame is a list or system identifying the members of the population from which the sample is selected.
Simple Random Sampling
In simple random sampling, every possible sample of the required size has an equal chance of being selected. Random numbers can be used to select individuals from a sampling frame.
Systematic Sampling
Systematic sampling selects members at regular intervals from a sampling frame. If the population contains N members and n are required, the sampling interval is approximately N/n.
A random starting point is normally chosen before applying the fixed interval.
Stratified Sampling
Stratified sampling divides the population into relevant strata and selects a sample from each stratum in proportion to the size of that stratum.
WORKED EXAMPLE // STRATIFIED SAMPLE
A college contains 600 Year 12 students and 400 Year 13 students. A sample of 50 students is required.
Year 12 sample = (600/1000) × 50 = 30
Year 13 sample = (400/1000) × 50 = 20
Therefore the stratified sample contains 30 Year 12 and 20 Year 13 students.
Quota Sampling
Quota sampling sets required numbers for categories of individuals. The interviewer then selects individuals until each quota is filled. Unlike stratified random sampling, the selection within each category is not necessarily random.
Opportunity Sampling
Opportunity sampling selects individuals who are readily available. It is usually convenient but can produce substantial bias because the people available may differ systematically from the wider population.
Sampling Bias
Bias occurs when the method of collecting data systematically favours some members or characteristics of the population over others.
Non-response
Non-response occurs when selected individuals do not provide the required information. If non-responders differ from responders, the resulting sample may be biased.
Response Bias
Response bias can occur when individuals give inaccurate answers, perhaps because a question is sensitive, leading or difficult to understand.
Designing a Good Sample
- Define the population clearly.
- Use an appropriate sampling frame.
- Choose a sampling method suitable for the investigation.
- Reduce opportunities for selection bias.
- Use an adequate sample size.
- Consider non-response and measurement issues.
- Ensure the wording of questions is neutral.
TOPIC 02 // DATA REPRESENTATION
Types of Data and Statistical Diagrams
Qualitative Data
Qualitative data describes categories or qualities rather than numerical measurements. Examples include eye colour, transport type and brand.
Quantitative Data
Quantitative data is numerical. It can be divided into discrete and continuous data.
Discrete Data
Discrete data can take separate, countable values. Examples include number of children in a family and number of defective components.
Continuous Data
Continuous data can take any value within an interval. Measurements such as height, mass, time and temperature are commonly continuous.
Primary and Secondary Data
- Primary data: collected directly for the investigation.
- Secondary data: obtained from an existing source.
Frequency Tables
A frequency table records how many observations occur in each category or class.
Grouped Data
When there are many numerical observations, values can be grouped into class intervals. Grouping makes data easier to display but loses some information because the exact values are no longer known.
Class Boundaries
For continuous measurements rounded to a stated accuracy, class boundaries may be needed. For example, values recorded to the nearest integer as 10, 11, 12 may correspond to boundaries 9.5, 10.5, 11.5 and so on.
Histograms
A histogram is used for continuous grouped data. The area of each bar is proportional to the frequency.
Therefore:
Stem-and-Leaf Diagrams
A stem-and-leaf diagram preserves the original observations while displaying their distribution. A key is essential.
EXAMPLE
A key such as 3 | 7 = 37 tells the reader how to interpret the diagram.
Scatter Diagrams
A scatter diagram displays paired observations for two variables. It can be used to investigate whether a linear relationship appears to exist.
Outliers
An outlier is an observation that lies unusually far from the main body of the data. An outlier should not automatically be deleted. Its cause should be investigated.
Interpreting Diagrams
When interpreting a diagram, consider centre, spread, shape, skewness, clusters, gaps and unusual observations.
TOPIC 03 // LOCATION AND SPREAD
Measures of Location and Variation
Arithmetic Mean
The mean uses every observation and is therefore sensitive to extreme values.
Mean from a Frequency Table
For grouped data, class midpoints are commonly used to estimate the mean.
Median
The median is the central observation after the data has been arranged in order. It is less affected by extreme observations than the mean.
Mode
The mode is the value or category occurring most frequently.
Range
The range measures the total spread but depends only on two observations.
Quartiles
The lower quartile Q₁ divides approximately the lowest quarter of the ordered data from the upper three quarters. Q₃ similarly separates the upper quarter.
Interquartile Range
The IQR measures the spread of the middle 50% of the data and is less affected by extreme values than the range.
Variance
For a dataset, variance measures the average squared distance from the mean. Its units are the square of the original units.
Standard Deviation
Standard deviation is expressed in the same units as the original data and gives a measure of typical spread around the mean.
Effect of Transformations
If every observation is increased by a constant a, the mean increases by a but the variance and standard deviation remain unchanged.
If every observation is multiplied by a constant b, the mean is multiplied by b, the variance is multiplied by b², and the standard deviation is multiplied by |b|.
Coding
Coding can simplify calculations by replacing x with a transformed variable such as y = (x − a)/b. The original statistics can then be recovered using the transformation.
WORKED EXAMPLE // CODING
Suppose y = (x − 50)/10 and the mean of y is 2.4.
x = 10y + 50
Therefore:
mean(x) = 10(2.4) + 50
mean(x) = 74
TOPIC 04 // PROBABILITY FIELD
Probability and Events
Probability provides a numerical measure of how likely an event is to occur.
A probability of 0 represents an impossible event and a probability of 1 represents a certain event.
Sample Space
The sample space is the set of all possible outcomes of an experiment.
Complement
Union
A ∪ B means that A or B or both occur.
Intersection
A ∩ B means that both A and B occur.
Addition Rule
The intersection is subtracted because it is counted twice when P(A) and P(B) are added.
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur together.
Therefore:
Conditional Probability
The notation P(A|B) means the probability of A occurring given that B has already occurred.
Independent Events
Events A and B are independent if the occurrence of one does not alter the probability of the other.
For independent events:
Tree Diagrams
Tree diagrams can be used to represent sequences of events. Probabilities along a path are multiplied, while probabilities of alternative paths are added.
Venn Diagrams
Venn diagrams are useful for visualising unions, intersections, complements and mutually exclusive events.
TOPIC 05 // DISCRETE RANDOM VARIABLES
Discrete Random Variables and Probability Distributions
A random variable assigns a numerical value to the outcome of a random experiment.
Discrete Random Variable
A discrete random variable can take a countable set of possible values. For example, X may represent the number of defective items in a sample.
Probability Distribution
A probability distribution lists the possible values of X together with their probabilities.
Every probability must satisfy:
Finding an Unknown Probability
If a distribution contains an unknown constant k, use the fact that the probabilities must add to 1.
WORKED EXAMPLE
Suppose probabilities are k, 2k, 3k and 4k.
k + 2k + 3k + 4k = 1
10k = 1
k = 0.1
Expected Value
The expected value is the long-run average value of X over a large number of repetitions of the experiment.
Expected Value of a Function
For suitable functions of X:
Variance
Standard Deviation
Linear Transformations
Adding a constant changes location but not spread. Multiplication changes both location and spread.
TOPIC 06 // BINOMIAL ENGINE
Binomial Distribution
The binomial distribution models the number of successes in a fixed number of independent trials when each trial has two outcomes and the probability of success remains constant.
The Four Conditions
- There is a fixed number n of trials.
- Each trial has exactly two relevant outcomes: success or failure.
- The probability of success p is constant.
- The trials are independent.
Probability of Exactly r Successes
Combination Notation
At Least
“At least r” means r or more.
Often the complement is easier:
At Most
More Than
Less Than
Mean
Variance
Standard Deviation
WORKED EXAMPLE // BINOMIAL
A machine produces components with probability 0.1 of being defective. Ten components are selected independently. Let X be the number of defective components.
X ~ B(10,0.1)
P(X=2)=C(10,2)(0.1)²(0.9)⁸
E(X)=np=10(0.1)=1
Var(X)=np(1-p)=10(0.1)(0.9)=0.9
When Binomial Is Not Appropriate
- The number of trials is not fixed.
- There are more than two relevant outcomes.
- The probability of success changes from trial to trial.
- The trials are not independent.
TOPIC 07 // CORRELATION AND REGRESSION
Correlation, Product Moment and Regression
Bivariate Data
Bivariate data consists of paired observations for two variables. For example, height and mass measured for the same group of people.
Scatter Diagram
A scatter diagram can reveal the direction, strength and approximate form of a relationship between two variables.
Positive Correlation
As one variable increases, the other tends to increase.
Negative Correlation
As one variable increases, the other tends to decrease.
No Linear Correlation
There may be little evidence of a linear relationship. This does not necessarily mean that no relationship of any kind exists.
Product Moment Correlation Coefficient
The product moment correlation coefficient, commonly denoted by r, measures the strength and direction of linear association.
Values close to 1 indicate strong positive linear correlation. Values close to −1 indicate strong negative linear correlation. Values close to 0 indicate weak linear correlation.
Regression Line
A regression line provides a model for predicting one variable from another.
Here a is the intercept and b is the gradient.
Interpolation
Interpolation predicts a value within the range of observed x-values. It is generally more defensible than extrapolation because it remains within the observed region.
Extrapolation
Extrapolation predicts beyond the observed range. The linear model may not remain valid outside the available data.
Residuals
A residual is the difference between an observed value and the value predicted by the regression model.
Interpreting r
The sign of r gives the direction of linear correlation, while the magnitude of r indicates its strength.
Correlation and Causation
A strong correlation can arise because one variable causes another, because both are influenced by another variable, because of coincidence, or because of the way the data was collected.
EXAM SYSTEM // ESSENTIAL TECHNIQUE
How to Approach S1 Questions
1. Identify the Statistical Structure
Before calculating, identify whether the question involves sampling, summary statistics, probability, a random variable, binomial modelling, or correlation/regression.
2. Define Variables
When using a probability model, define X clearly. For example: “Let X be the number of defective components.”
3. State the Model
For binomial questions write:
This demonstrates that you have identified the correct distribution.
4. Keep Full Precision
Do not round intermediate values unnecessarily. Round only at the appropriate stage, normally at the final answer.
5. Interpret in Context
A statistical answer should often be interpreted using the original context. Do not leave an answer as a disconnected numerical value when the question asks for a conclusion.
6. Check Plausibility
- Probability must lie between 0 and 1.
- Standard deviation cannot be negative.
- Variance cannot be negative.
- A probability distribution must sum to 1.
- A correlation coefficient must lie between −1 and 1.
- Expected values do not necessarily have to be possible individual outcomes.
SYSTEM ARCHIVE // FORMULA DATABASE
S1 Formula Archive
FINAL EXAM // 100 QUESTION S1 ASSESSMENT
Statistics 1 — 100 Question Examination
This examination is designed as a comprehensive revision system. Attempt all questions before opening the answers. Questions progress from foundational knowledge to calculation, interpretation and multi-step statistical reasoning.
Try to complete the entire paper without opening the answer sections. Use a calculator where appropriate and show working for calculation questions.
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The complete set of individuals or objects about which the investigation is concerned.
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A subset of the population selected for observation.
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Information is collected from every member of the population, so sampling variation is avoided.
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It can be expensive, time-consuming or impractical for a large population.
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A list or system identifying members of the population from which a sample can be selected.
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Every possible sample of the required size has an equal chance of being selected.
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800 ÷ 40 = 20.
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(600/1000) × 50 = 30 boys.
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The people who are easiest to access may differ systematically from the wider population.
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Bias arising when people who do not respond differ in relevant characteristics from those who respond.
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Discrete data has countable separate values; continuous data can take any value within an interval.
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Continuous.
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Discrete.
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Frequency.
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30 ÷ 5 = 6.
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4 × 8 = 32.
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The exact values within each class are unknown, so class midpoints are used to represent the observations.
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Minimum, lower quartile, median, upper quartile and maximum.
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An observation unusually far from the main body of the data.
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For example, centre and spread; other valid answers include skewness, clusters, gaps and outliers.
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(4+7+9+10+15)/5 = 9.
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Ordered data: 3,4,7,8,12. Median = 7.
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21 − 7 = 14.
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31 − 14 = 17.
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The median is less affected by extreme values.
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Σfx = 6+8+40 = 54; Σf = 10. Mean = 5.4.
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29 − 5² = 4.
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√49 = 7.
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It remains unchanged.
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The variance is multiplied by 3² = 9.
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0 ≤ P(A) ≤ 1.
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1 − 0.38 = 0.62.
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A and B both occur.
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A or B or both occur.
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P(A∪B)=P(A)+P(B)−P(A∩B).
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0.
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0.4 + 0.3 = 0.7.
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0.6 × 0.5 = 0.3.
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P(A|B)=P(A∩B)/P(B).
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0.18 ÷ 0.3 = 0.6.
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A random variable that takes a countable set of possible values.
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1.
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10k=1, so k=0.1.
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E(X)=ΣxP(X=x).
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1(0.2)+2(0.5)+4(0.3)=2.4.
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1²(0.2)+2²(0.5)+4²(0.3)=0.2+2+4.8=7.
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7−2.4² = 1.24.
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√1.24 ≈ 1.114.
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2(4)+5 = 13.
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3²×4 = 36.
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Fixed number of trials, two outcomes, constant probability of success and independent trials.
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X~B(20,0.4).
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10×0.2 = 2.
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10×0.2×0.8 = 1.6.
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(0.8)¹⁰ ≈ 0.1074.
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C(10,3)(0.2)³(0.8)⁷.
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X ≥ 4.
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X ≤ 4.
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P(X≥5)=1−P(X≤4).
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For example, when the probability of success changes between trials.
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X = number of heads. X~B(8,0.7).
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C(8,5)(0.7)⁵(0.3)³ ≈ 0.2541.
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P(X=7)+P(X=8) ≈ 0.2553.
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8×0.7 = 5.6.
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8×0.7×0.3 = 1.68.
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0.95²⁰ ≈ 0.3585.
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20×0.05 = 1.
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20×0.05×0.95 = 0.95.
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The binomial distribution assumes each trial has the same probability p of success.
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The outcome of one trial must not affect the probability of the outcome of another trial.
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As one variable increases, the other tends to increase.
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As one variable increases, the other tends to decrease.
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−1 ≤ r ≤ 1.
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There is a strong positive linear correlation.
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There is a strong negative linear correlation.
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There is little or no linear correlation. It does not prove that there is no relationship at all.
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y=a+bx.
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The gradient is 3.
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12+3(5)=27.
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Interpolation predicts within the observed range; extrapolation predicts outside it.
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27−24 = 3.
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The relationship observed in the available data may not continue outside the observed range.
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No.
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A third variable may influence both variables.
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5+24 = 29.
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38−42 = −4.
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It indicates the direction of linear association: positive or negative.
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It indicates the strength of the linear relationship.
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It allows the relationship, possible outliers and suitability of a linear model to be assessed visually.
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The prediction is an extrapolation and the linear relationship may not continue.
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1200÷60 = 20.
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4(7)−3 = 25.
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4²×5 = 80.
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15×0.4 = 6.
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15×0.4×0.6 = 3.6.
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P(X≥13)=1−P(X≤12).
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0.15+0.25+k+0.30=1, so k=0.30.
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0(0.2)+1(0.5)+2(0.3)=1.1.
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Mean = 3(20)+5 = 65. Standard deviation = 3(4) = 12.
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Let X be the number of defective components.
X~B(25,0.08)
E(X)=25(0.08)=2
Var(X)=25(0.08)(0.92)=1.84
P(X≥3)=1−P(X≤2)
So:
P(X≥3)=1−[P(X=0)+P(X=1)+P(X=2)]
SYSTEM CHECK // S1 MASTERY
S1 Mastery Checklist
- I can distinguish a population from a sample.
- I can explain census and sampling frames.
- I can compare simple random, systematic, stratified, quota and opportunity sampling.
- I can identify potential sources of sampling bias.
- I can distinguish discrete and continuous data.
- I can interpret frequency tables, histograms, stem-and-leaf diagrams, box plots and scatter diagrams.
- I understand frequency density.
- I can calculate and interpret mean, median, mode, range, quartiles and IQR.
- I can calculate variance and standard deviation.
- I understand the effects of linear transformations on mean and variance.
- I can use complements, unions and intersections.
- I can distinguish mutually exclusive and independent events.
- I can calculate conditional probabilities.
- I can construct and interpret tree diagrams.
- I can construct probability distributions.
- I can calculate expected value and variance of a discrete random variable.
- I can recognise and use the binomial distribution.
- I can calculate exact, cumulative, at-least and at-most binomial probabilities.
- I can calculate the mean and variance of a binomial distribution.
- I can interpret scatter diagrams and correlation coefficients.
- I understand the difference between correlation and causation.
- I can use regression equations for prediction.
- I understand interpolation and extrapolation.
- I can calculate and interpret residuals.
- I can complete a full 100-question S1 assessment.
Do not judge mastery by whether you can remember the formulae. A strong S1 student should be able to identify the correct statistical model, explain why it applies, perform the calculation accurately, interpret the result and recognise when a method is inappropriate.