## Sampling and Data Collection

When conducting statistical investigations, we often collect data from a sample rather than the entire population. The population is the complete set of individuals or items we are interested in, while a sample is a smaller, representative subset. A census involves collecting data from every member of the population. Sampling is usually preferred due to cost, time, and practicality.

## Sampling Methods

Choosing the right sampling method is crucial to ensure the sample is representative and avoids bias.

Random Sampling (Probability Sampling)

These methods give every member of the population a known chance of being selected, reducing bias.

• Simple Random Sampling: Every member has an equal chance. Requires a sampling frame (a list of all members).
• *Advantages:* Unbiased, easy to understand.
• *Disadvantages:* Can be impractical for large populations.
• Systematic Sampling: Selects items at regular intervals from an ordered list (e.g., every 10th person).
• *Advantages:* Simple to implement.
• *Disadvantages:* Can be biased if there's a pattern in the list.
• Stratified Sampling: Divides the population into distinct subgroups (strata) and then takes a random sample from each stratum, proportional to its size in the population.
• *Advantages:* Ensures representation from key subgroups, more precise.
• *Disadvantages:* Requires knowledge of population characteristics, more complex.

Non-Random Sampling (Non-Probability Sampling)

These methods do not give every member a known chance of selection and are prone to bias, making results less generalisable.

• Opportunity/Convenience Sampling: Selecting individuals who are readily available.
• *Advantages:* Quick, easy, inexpensive.
• *Disadvantages:* Highly biased, not representative.
• Quota Sampling: Interviewers select individuals until a pre-set quota for specific characteristics (e.g., age, gender) is met. Selection within quotas is non-random.
• *Advantages:* Quick, no sampling frame needed.
• *Disadvantages:* Interviewer bias, not truly random.

## Data Collection

Primary vs. Secondary Data

• Primary Data: Data collected directly by the researcher for the specific investigation (e.g., through surveys, experiments).
• *Pros:* Specific to needs, control over quality.
• *Cons:* Time-consuming, expensive.
• Secondary Data: Data collected by someone else for a different purpose, then used by the researcher (e.g., government statistics, published reports).
• *Pros:* Quick, cheap, readily available.
• *Cons:* May not fit exact needs, quality/reliability can be unknown.

Questionnaire Design

Well-designed questionnaires are vital for accurate data.

• Open Questions: Allow free, detailed responses (qualitative data).
• Closed Questions: Provide fixed response options (quantitative data), easier to analyse.
• Good questionnaires avoid leading questions, are clear, unambiguous, and logically structured. Pilot testing helps refine them.

## Tables, Charts and Diagrams

Tables, charts, and diagrams are essential tools for organising, summarising, and presenting data effectively. Choosing the correct display depends on the type of data you are working with.

## Types of Data

• Qualitative (Categorical) Data: Non-numerical data that can be sorted into categories (e.g., colours, types of transport).
• Quantitative Data: Numerical data.
• Discrete Data: Can only take specific, fixed values, often whole numbers (e.g., number of siblings, shoe size).
• Continuous Data: Can take any value within a given range (e.g., height, weight, time).

## Tables

• Frequency Tables: List categories or values and their corresponding frequencies.
• Grouped Frequency Tables: Used for large ranges of discrete or continuous data, grouping values into class intervals. Make sure class intervals are non-overlapping and cover all data.
• Two-way Tables: Display the frequencies of two categorical variables simultaneously, showing relationships between them.

## Charts for Categorical and Discrete Data

• Bar Charts: Use separate bars of equal width to represent the frequency of each category or discrete value. The height of the bar is proportional to the frequency. There are gaps between bars.
• Pie Charts: Represent parts of a whole. The circle is divided into sectors, where the angle of each sector is proportional to its frequency. Angle = (Frequency / Total Frequency) * 360°.
• Pictograms: Use symbols to represent frequencies. A key is essential to show what each symbol represents.

## Charts for Continuous Data

• Histograms: Used for continuous data, especially when class intervals are unequal. The area of each bar, not its height, is proportional to the frequency.
• Frequency Density = Frequency / Class Width. This is plotted on the y-axis. There are no gaps between bars.
• Frequency Polygons: Plotted by marking the midpoint of the top of each bar in a histogram (or the midpoint of each class interval for grouped frequency data) and joining these points with straight lines.
• Cumulative Frequency Graphs: Plotted using the upper class boundaries against cumulative frequency. Used to estimate the median, quartiles (lower, upper), interquartile range (IQR), and percentiles.

## Scatter Graphs

• Used to display bivariate data (two variables) to show if there is a correlation (relationship) between them. Points are plotted on a coordinate grid. Correlation can be positive, negative, or no correlation.

## Misleading Diagrams

Be aware of how diagrams can be misleading:

• Broken or inconsistent scales on axes.
• Unequal bar widths in bar charts.
• Missing labels or titles.
• Using 3D effects that distort proportions.
• Starting the y-axis at a value other than zero to exaggerate differences.

## Measures of Central Tendency and Spread

This topic explores ways to summarise and compare sets of data using single values. Understanding these measures is crucial for interpreting data distributions.

Measures of Central Tendency

These describe the 'average' or 'typical' value in a dataset.

• Mean: Calculated by summing all values and dividing by the number of values. It uses all data points but is sensitive to outliers (extreme values). The mean is best suited for symmetrical data distributions.
• Median: The middle value when data is ordered from smallest to largest. If there's an even number of values, it's the average of the two middle values. The median is not affected by outliers and is best for skewed data distributions.
• Mode: The value that appears most frequently in a dataset. There can be no mode, one mode, or multiple modes (e.g., bimodal). It's the only measure of central tendency suitable for categorical data.

Choosing the best measure of central tendency:

• Use the mean for symmetrical data without outliers.
• Use the median for skewed data or data with outliers.
• Use the mode for categorical data or when identifying the most popular item.

Measures of Spread (Dispersion)

These describe how spread out or varied the data points are.

• Range: The difference between the highest and lowest values in a dataset. It's easy to calculate but heavily influenced by outliers, making it a less robust measure of spread.
• Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1) (IQR = Q3 - Q1). Q1 is the median of the lower half of the data, and Q3 is the median of the upper half. The IQR represents the spread of the middle 50% of the data and is much less affected by outliers than the range. It is best for skewed data.
• Standard Deviation: Measures the average distance of each data point from the mean. A larger standard deviation indicates greater variability or spread in the data. It is the most robust measure of spread for symmetrical data and is often calculated using a calculator in GCSE.

Choosing the best measure of spread:

• Use the range for a quick, simple overview, but be aware of its sensitivity to outliers.
• Use the IQR for skewed data or data with outliers, as it focuses on the central spread.
• Use standard deviation with the mean for symmetrical data, providing a comprehensive measure of spread.

## Introduction to Probability

Probability measures the likelihood of an event occurring. It is always a value between 0 and 1 (inclusive), or 0% and 100%.

• 0 means impossible.
• 1 means certain.
• 0.5 (or 1/2) means an even chance.

The sample space is the set of all possible outcomes of an experiment. The sum of probabilities for all possible outcomes in a sample space must equal 1.

Notation: P(A) represents the probability of event A happening.

## Types of Probability

• Theoretical Probability: Based on reasoning and equally likely outcomes. Calculated as: (Number of favourable outcomes) / (Total number of possible outcomes).
• Experimental Probability (Relative Frequency): Based on observations from an experiment. Calculated as: (Number of times an event occurs) / (Total number of trials). As the number of trials increases, experimental probability tends to get closer to theoretical probability.

## Key Probability Rules

• Complementary Events: The event that A does *not* happen is denoted as A' (or Aᶜ). P(A') = 1 - P(A).
• Mutually Exclusive Events: Two events are mutually exclusive if they cannot happen at the same time (e.g., rolling a 1 and a 2 on a single die). For mutually exclusive events A and B: P(A or B) = P(A) + P(B).
• Independent Events: Two events are independent if the outcome of one does not affect the outcome of the other (e.g., flipping a coin twice). For independent events A and B: P(A and B) = P(A) × P(B).
• Conditional Probability: The probability of event A happening *given* that event B has already happened is denoted P(A|B). It means we are considering a reduced sample space where B has occurred. Often calculated using tree diagrams or two-way tables.

## Visualising Probability

• Tree Diagrams: Used for sequences of events. Probabilities along branches are multiplied for "AND" outcomes. Probabilities at the end of branches are added for "OR" outcomes. Useful for dependent events (without replacement) and independent events.
• Venn Diagrams: Represent events as circles within a rectangle (the sample space). Overlapping regions show the intersection (A and B). The combined area of circles shows the union (A or B).
• Two-way Tables: Organise data by two categorical variables, making it easy to find probabilities of combined events or conditional probabilities.

## Expected Outcomes

The expected number of outcomes for an event in a given number of trials is calculated as: P(event) × Number of trials.

## Correlation and Regression

Correlation describes the strength and direction of a linear relationship between two variables. When plotting data on a scatter diagram, we look for a trend.

• Positive correlation: As one variable increases, the other tends to increase. The points generally go upwards from left to right.
• Negative correlation: As one variable increases, the other tends to decrease. The points generally go downwards from left to right.
• No correlation: There is no clear linear pattern between the variables.
• The strength of correlation can be described as strong, moderate, or weak, depending on how closely the points cluster around a straight line. The closer they are, the stronger the correlation.
• It's crucial to remember that correlation does not imply causation. Just because two variables are correlated, it doesn't mean one causes the other; there might be other influencing factors (lurking variables).

A scatter diagram is a graph used to display the relationship between two quantitative variables. Each point on the diagram represents a pair of values for the two variables. It helps visualise the correlation.

Regression involves drawing a line of best fit on a scatter diagram. This straight line represents the general trend of the data and can be used to model the relationship.

• To draw a line of best fit by eye, ensure it follows the trend of the data, has roughly an equal number of points above and below it, and ideally passes through the mean point (mean of x, mean of y).
• The equation of the line of best fit is typically given as y = a + bx, where 'a' is the y-intercept (value of y when x=0) and 'b' is the gradient (change in y for a unit change in x). You should be able to interpret these in context.
• Interpolation is making predictions within the range of the observed data. These predictions are generally more reliable.
• Extrapolation is making predictions outside the range of the observed data. These predictions are generally less reliable because the trend might not continue beyond the observed data range.
• The reliability of predictions depends on the strength of the correlation (stronger correlation means more reliable predictions) and whether you are interpolating or extrapolating. Outliers can also affect the line of best fit and the reliability of predictions.

## Index Numbers

Index numbers are a way to measure changes in a variable (like price or quantity) over time, relative to a base period. They simplify comparisons and highlight trends. The base period is usually set to an index of 100. To calculate an index number:

Index Number = (Current Value / Base Value) * 100.

For example, if the price of an item was £10 in the base year and £12 now, the index number is (12/10)*100 = 120, indicating a 20% increase.

## Time Series Components

A time series is a sequence of data points collected at regular intervals over time. It can be broken down into four main components:

• Trend: The long-term general direction or movement of the data (e.g., increasing, decreasing, or stable).
• Seasonal Variation: Regular, predictable patterns that repeat over a fixed period, such as daily, weekly, monthly, or yearly. These are often linked to seasons, holidays, or recurring events.
• Cyclical Variation: Longer-term patterns that are not fixed in duration, often linked to economic cycles (e.g., boom and recession). These are less predictable than seasonal variations.
• Random Variation (or Residual Variation): Irregular, unpredictable fluctuations in the data caused by chance or unusual, one-off events.

## Moving Averages

Moving averages are used to smooth out short-term fluctuations in time series data, making the underlying trend easier to identify. A moving average is calculated by averaging a fixed number of consecutive data points. For example, a 3-point moving average sums three consecutive data points and divides by three. When using an even number of data points (e.g., a 4-point moving average), the result needs to be centred by averaging two consecutive moving averages to align it with the original time point.

## Seasonal Adjustment and Forecasting

Seasonal adjustment removes the effect of seasonal variation from data, allowing the underlying trend to be seen more clearly. This involves calculating seasonal differences (Original Data - Moving Average) for each period, then finding the average seasonal difference for each specific period (e.g., average for all Q1s). Deseasonalised data is then obtained by subtracting the average seasonal difference from the original data.

Forecasting involves predicting future values. This is typically done by extending the identified trend (e.g., using a trend line equation) and then re-applying the appropriate average seasonal difference for the forecast period. For example, Forecast = Trend Value + Average Seasonal Difference.