## Sampling and Data Collection

Population and Samples

In statistics, the population refers to the entire group of individuals or items that a researcher is interested in studying. It's often impractical, too costly, or too time-consuming to collect data from every member of a population. Therefore, a sample – a smaller, representative subset of the population – is often studied instead. A sampling frame is a list of all members of the population from which a sample can be drawn (e.g., a school register, a customer database).

## Sampling Methods

Random Sampling (Probability Sampling): Each member of the population has a known, non-zero chance of being selected, aiming for an unbiased and representative sample.

• Simple Random Sampling: Every member of the population has an equal chance of being selected. Methods include drawing names from a hat or using a random number generator. It's unbiased but requires a complete sampling frame.
• Systematic Sampling: Selecting every nth item from a sampling frame after a random starting point. It's simple to implement but can be biased if there's a pattern in the list that aligns with the sampling interval.
• Stratified Sampling: The population is divided into distinct subgroups called strata (e.g., age groups, genders). A simple random sample is then taken from each stratum, proportional to its size in the population. This ensures representation from all key subgroups and is highly representative.

Non-Random Sampling (Non-Probability Sampling): Selection is not based on chance, meaning some members have no chance of selection, potentially leading to bias.

• Opportunity (Convenience) Sampling: Selecting individuals who are readily available or easiest to reach. It's quick and inexpensive but often highly biased and not representative.
• Quota Sampling: Similar to stratified sampling, but interviewers are given a quota to fill for different subgroups. Selection within the subgroups is non-random, often left to the interviewer, which can introduce bias.

## Types of Data

• Primary Data: Data collected firsthand by the researcher for a specific purpose (e.g., surveys, experiments). It's reliable and specific but can be time-consuming and costly.
• Secondary Data: Data that has already been collected by someone else for a different purpose (e.g., government statistics, published research). It's quick and cheap to access but may not perfectly fit the research needs and its reliability needs careful checking.

• Qualitative Data: Non-numerical, descriptive data that explores opinions, characteristics, or categories (e.g., favourite colour, reasons for choosing a product).
• Quantitative Data: Numerical data that can be measured or counted.
• Discrete Data: Can only take specific, distinct values, often counted (e.g., number of siblings, shoe size).
• Continuous Data: Can take any value within a given range, often measured (e.g., height, weight, temperature).

## Data Collection Considerations

When collecting data, especially through questionnaires, it's crucial to avoid bias. Questions should be clear, unambiguous, and not leading (suggesting a preferred answer). A pilot study can help identify issues before full data collection.

## Tables, Charts and Diagrams

Visualising data is crucial in statistics to identify patterns, trends, and anomalies more easily than looking at raw numbers. Choosing the correct chart depends on the type of data you have.

Choosing the Right Chart

• Categorical Data: Represents categories or groups (e.g., colours, types of fruit). Use Bar Charts to show frequencies or counts, or Pie Charts to show proportions of a whole.
• Numerical Data: Represents quantities.
• Discrete Data: Can only take specific values (e.g., number of siblings).
• Continuous Data: Can take any value within a range (e.g., height, time).

Common Charts and Their Uses

• Bar Charts: Display frequencies for categorical or discrete data. Bars are separated by gaps. Dual bar charts compare two sets of categories.
• Pie Charts: Show proportions of a whole for categorical data. Each slice's angle is proportional to its frequency (Angle = (Frequency / Total Frequency) * 360°).
• Histograms: Used for grouped continuous data. Bars have no gaps, and the area of each bar represents the frequency. For unequal class widths, the y-axis is frequency density (Frequency / Class Width).
• Frequency Polygons: Connect the midpoints of the tops of histogram bars, or plot midpoints against frequency, for grouped continuous data.
• Line Graphs: Show how a variable changes over time, highlighting trends.
• Scatter Graphs: Display the relationship between two numerical variables. Used to observe correlation (positive, negative, or no correlation). A line of best fit can be drawn to estimate values (interpolation) or predict beyond the data (extrapolation).
• Cumulative Frequency Graphs: Plot cumulative frequency against the upper class boundary for continuous data. Used to estimate the median, quartiles, and interquartile range (IQR).
• Box Plots (Box and Whisker Diagrams): Summarise the distribution of continuous data using five key values: minimum, lower quartile (Q1), median (Q2), upper quartile (Q3), and maximum. Useful for comparing distributions and identifying outliers.
• Stem and Leaf Diagrams: Organise discrete or continuous data, preserving individual data values. They show the shape of the distribution and are useful for finding the median and mode. Back-to-back stem and leaf diagrams compare two datasets.

Key Features of Good Charts

All charts should have a clear title, properly labelled axes with units, and an appropriate scale. A key or legend may be needed.

Misleading Diagrams

Charts can be manipulated to misrepresent data. Be aware of:

• Truncated y-axis: Starting the y-axis above zero can exaggerate differences.
• Inconsistent scales: Uneven intervals on axes.
• 3D effects: Can distort perception of values.
• Inappropriate chart type: Using a pie chart for non-proportional data.

## Measures of Central Tendency

Measures of central tendency describe the centre or average of a dataset.

• Mean: The arithmetic average. Calculated by summing all values and dividing by the number of values.
• Advantages: Uses all data values in its calculation, making it a good representation for symmetrical data.
• Disadvantages: Heavily affected by outliers (extreme values) and skewed distributions.
• Median: The middle value when the data is arranged in order (ascending or descending). If there's an even number of data points, it's the average of the two middle values.
• Advantages: Not affected by outliers or skewed distributions, as it only considers the position of values.
• Disadvantages: Does not use all data values in its calculation.
• Mode: The most frequently occurring value in a dataset. A dataset can have one mode (unimodal), multiple modes (multimodal), or no mode.
• Advantages: Can be used for categorical data (non-numerical), easy to identify. Not affected by outliers.
• Disadvantages: May not exist, or there might be multiple modes, making it less useful for numerical data. Doesn't use all data values.

Choosing a Measure of Central Tendency:

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

## Measures of Spread

Measures of spread (or dispersion) describe how spread out or variable the data is.

• Range: The difference between the highest and lowest values in a dataset.
• Advantages: Easy to calculate and understand.
• Disadvantages: Heavily affected by outliers, as it only uses the two extreme values.
• Interquartile Range (IQR): The difference between the upper quartile (Q3) and the lower quartile (Q1). It represents the spread of the middle 50% of the data.
• To find Q1: The median of the lower half of the data.
• To find Q3: The median of the upper half of the data.
• Advantages: Not affected by outliers, as it excludes the extreme 25% of data at each end.
• Disadvantages: Does not use all data values.
• Standard Deviation: A measure of the average distance of each data point from the mean. A low standard deviation indicates data points are close to the mean, while a high standard deviation indicates data points are spread out over a wider range of values.
• Advantages: Uses all data values, useful for normal distributions.
• Disadvantages: Affected by outliers, more complex to calculate (though often done by calculator in GCSE).

Choosing a Measure of Spread:

• Use the range for a quick, simple idea of spread, but be aware of outliers.
• Use the IQR when the median is used for central tendency (i.e., for skewed data or with outliers).
• Use the standard deviation when the mean is used for central tendency (i.e., for symmetrical data without outliers).

## Understanding Probability

Probability is a measure of the likelihood that an event will occur. It is expressed as a value between 0 and 1, inclusive.

• 0 means the event is impossible.
• 1 means the event is certain.
• Probabilities can be written as fractions, decimals, or percentages.
• The sum of probabilities of all possible outcomes for an event must equal 1.
• The probability of an event not happening is called its complement. If P(A) is the probability of event A happening, then P(A') = 1 - P(A) is the probability of A not happening.

## Types of Probability

• Theoretical Probability: Calculated based on reasoning and the assumption that all outcomes are equally likely.

P(Event) = (Number of favourable outcomes) / (Total number of possible outcomes).

• Experimental Probability (Relative Frequency): Calculated from observed data after conducting trials or experiments.

P(Event) = (Number of times the event occurred) / (Total number of trials).

As the number of trials increases, experimental probability tends to get closer to theoretical probability.

## Combining Events

• Mutually Exclusive Events: Events that cannot happen at the same time.

If A and B are mutually exclusive, P(A or B) = P(A) + P(B).

• Independent Events: Events where the outcome of one does not affect the outcome of the other.

If A and B are independent, P(A and B) = P(A) × P(B).

• Dependent Events: Events where the outcome of the first event *does* affect the probability of the second event (e.g., drawing cards without replacement). Tree diagrams are often used for these.

## Conditional Probability

Conditional probability is the probability of an event occurring given that another event has already occurred. It is denoted as P(A|B), which means "the probability of A given B".

• P(A|B) = P(A and B) / P(B) (where P(B) > 0).
• This formula is crucial for understanding how probabilities change based on new information.

## Tools for Probability

• Sample Space Diagrams: List all possible outcomes of an event, especially useful for two combined events (e.g., rolling two dice).
• Tree Diagrams: Visualise sequences of events, particularly useful for dependent events or multiple stages. Branches are labelled with probabilities, and outcomes are found by multiplying along branches.
• Venn Diagrams: Represent relationships between sets of events, showing overlaps (intersections) and unions.

## Correlation and Scatter Diagrams

Correlation describes the relationship between two variables. When you plot paired data on a scatter diagram, each point represents a pair of values. The pattern of these points helps visualise the correlation.

## Types and Strength of Correlation

• Positive Correlation: As one variable increases, the other variable also tends to increase. The points on the scatter diagram generally rise from left to right.
• Negative Correlation: As one variable increases, the other variable tends to decrease. The points generally fall from left to right.
• No Correlation: There is no clear pattern between the variables. The points are scattered randomly.

The strength of correlation indicates how closely the points cluster around a straight line.

• Strong Correlation: Points are very close to forming a straight line.
• Moderate Correlation: Points show a general trend but are more spread out.
• Weak Correlation: Points show a very loose trend, if any.

## The Correlation Coefficient (r)

The correlation coefficient (r) is a numerical measure of the strength and direction of a linear relationship.

• It ranges from -1 to +1.
• r = +1 indicates perfect strong positive correlation.
• r = -1 indicates perfect strong negative correlation.
• r = 0 indicates no linear correlation.
• Values closer to +1 or -1 mean stronger correlation. Values closer to 0 mean weaker correlation.

## Correlation vs. Causation

It is vital 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 a confounding variable or it could be a complete coincidence.

## Line of Best Fit (Regression Line)

For data showing a linear correlation, you can draw a line of best fit (also called a regression line) on a scatter diagram. This line should pass through the general trend of the points, with roughly an equal number of points above and below it.

• Interpolation: Using the line of best fit to make predictions *within* the range of the original data. This is generally reliable.
• Extrapolation: Using the line of best fit to make predictions *outside* the range of the original data. This is often unreliable and should be done with caution, as the trend might not continue.

## Outliers

Outliers are data points that lie far away from the general trend of the other points. They can significantly affect the correlation coefficient and the position of the line of best fit.

## Index Numbers

Index numbers are used to measure the percentage change in a variable over time, relative to a specific base period. The base period is always assigned an index of 100. They simplify comparisons and highlight trends.

Calculation:

Index Number = (Value in current period / Value in base period) * 100

For example, if the price of an item was £10 in the base year and £12 in the current year, the index number would be (12/10) * 100 = 120. This indicates a 20% increase.

Common examples include the Retail Price Index (RPI) and Consumer Price Index (CPI), which measure inflation by tracking changes in the prices of a 'basket' of goods and services. A key advantage is ease of comparison; a limitation is that the choice of base period or basket of goods might not be representative for everyone.

## Time Series

A time series is a sequence of data points measured at successive points in time. Analysing time series helps to identify patterns and make forecasts.

Components of a Time Series:

• Trend: The long-term general direction of the data (e.g., increasing, decreasing, stable).
• Seasonal Variation: Regular, predictable fluctuations that occur within a specific period (e.g., within a year, month, or week). For example, ice cream sales are higher in summer.
• Cyclical Variation: Longer-term fluctuations over several years, often linked to economic cycles (e.g., boom, recession). Less regular than seasonal variation.
• Random Variation (Residual): Unpredictable, irregular fluctuations due to chance events or unusual circumstances.

Moving Averages:

Moving averages are used to smooth out short-term fluctuations (seasonal and random variations) in a time series, revealing the underlying trend more clearly. A 3-point moving average, for instance, is the average of three consecutive data points.

For an even number of points (e.g., a 4-point moving average), the average is initially placed between data points. To align it with an actual time point, you must centre the moving average by taking the average of two consecutive moving averages.

Forecasting:

Forecasting involves using past data patterns to predict future values. This typically involves:

1. Calculating moving averages to determine the trend.

2. Calculating seasonal variations by subtracting the trend from the actual values (Actual - Trend).

3. Averaging the seasonal variations for each specific period (e.g., Quarter 1, Quarter 2).

4. Projecting the future trend line.

5. Adding the average seasonal variation to the projected trend for each future period.

The additive model for a time series is often expressed as: Time Series = Trend + Seasonal Variation + Random Variation. Limitations of forecasting include the assumption that past patterns will continue and the impact of unforeseen events.