Topic 6: Statistics

Statistics is the branch of maths that deals with collecting, organising, summarising and interpreting data. In the AQA GCSE you are expected not only to calculate averages and draw charts, but also to comment on what the numbers mean and to compare two sets of data sensibly. The skills here reward careful, accurate work: reading a scale correctly, choosing the right average, and writing a clear comparison sentence can each earn marks. This guide walks through every part of Topic 6 in the order you usually meet it, building from simple lists up to histograms and frequency density, which are among the most demanding ideas at this level. Throughout, focus on two things: getting the calculation right, and explaining the result in the context of the question.

Types of Data and Sampling

Data comes in different forms and knowing the type tells you which methods are sensible. Qualitative (categorical) data describes qualities, such as colour or favourite sport. Quantitative data is numerical and splits into two kinds: discrete data can only take particular values, usually whole numbers (number of pets, goals scored), while continuous data can take any value in a range and is measured rather than counted (height, mass, time). Continuous values are always rounded, so a height recorded as 170 cm really means somewhere in a small interval around 170. A population is the whole group you are interested in; a sample is a smaller part of it that you actually study. We sample because surveying an entire population is usually too slow, costly or impractical. The trade-off is that a sample gives only an estimate of the population, so the sample must be chosen well. A good sample is representative and large enough to be reliable, but not so large it becomes unmanageable. In simple random sampling every member of the population has an equal chance of being chosen. In practice you number every member and use random numbers (or a calculator or a hat) to pick the sample, which removes human choice and so removes one source of bias. In stratified sampling the population is split into groups called strata (for example by year group or by gender) and you sample from each stratum in proportion to its size. The number you take from a stratum is given by sample size $\times \frac{\text{stratum size}}{\text{population size}}$. For example, to take a stratified sample of 60 from a school of 900 where 300 are in Year 11, you sample $60\times\frac{300}{900}=20$ Year 11 students. Always round these to whole people and check the parts add up to the total. Bias is anything that makes the sample unrepresentative, so that results systematically lean one way. Bias creeps in through a poor sampling method (asking only your friends), a sample that is too small, an unrepresentative time or place (surveying a gym about exercise habits), or leading questions. To reduce bias, use a random or stratified method, a large enough sample, and neutral questions with non-overlapping response boxes.

Averages and Range from Lists

An average is a single value that represents a typical value of a data set. There are three averages you must know. The mean is the total of the values divided by how many there are: $\text{mean}=\frac{\text{sum of values}}{\text{number of values}}$. The median is the middle value once the data is placed in order; if there are two middle values you take their mean. The mode is the most common value, the one that occurs most often, and a data set can have no mode or more than one. For a list of $n$ ordered values the median is in position $\frac{n+1}{2}$. With nine values that is position $\frac{9+1}{2}=5$, the fifth value. With ten values position $\frac{10+1}{2}=5.5$ means halfway between the fifth and sixth values, so you average those two. Always sort the list first; forgetting to order the data is the most common median error. The range measures spread (how spread out the data is) and is the largest value minus the smallest value: $\text{range}=\text{max}-\text{min}$. A small range means the data is consistent and clustered; a large range means it is varied. The range is not an average; it tells you about variation, not about a typical value. Each average has strengths. The mean uses every value, which makes it sensitive to extreme values called outliers: one very large value drags the mean up. The median ignores how extreme the outliers are and so is more resistant to them, making it a better choice for skewed data such as house prices or salaries. The mode is the only average you can use for qualitative data such as favourite colour. Knowing which to recommend, and why, is often worth a mark.

Averages from Frequency Tables

A frequency table records how many times each value occurs, which saves listing repeated values. Suppose a column $x$ holds the values and $f$ holds their frequencies. The mode is simply the value with the highest frequency. The total number of data items is $\sum f$, the sum of the frequency column. To find the mean you must account for how often each value appears, so you add a column for $fx$, the value multiplied by its frequency. The mean is then $\bar{x}=\frac{\sum fx}{\sum f}$, the total of the $fx$ column divided by the total frequency. For example, if a value of 3 occurs 5 times it contributes $3\times 5=15$ to the numerator and 5 to the denominator. A frequent slip is to divide by the number of different values instead of by $\sum f$; always divide by the total frequency. The median from a frequency table is found by locating the $\frac{n+1}{2}$th item, where $n=\sum f$. Build up a running total of the frequencies (the cumulative frequency) and find which value the target position lands in. If $n=40$ the median is between the 20th and 21st items; read down the running totals until you pass position 20.5, and that value is the median. These tables sometimes appear with the data already sorted because $x$ is listed in order, but check before reading off the median. A clear, well-labelled extra column for $fx$ keeps the working tidy and helps the examiner award method marks even if your final arithmetic slips.

Grouped Data: Estimated Mean and Modal Class

When data is grouped into class intervals (for example heights from 150 to 160 cm) you no longer know the individual values, only how many fall in each class. Because the exact values are lost, you cannot find the true mean, only an estimate. You assume every value in a class sits at the class midpoint, the value halfway between the lower and upper bounds. For the class $150 \le h \lt 160$ the midpoint is $\frac{150+160}{2}=155$. To estimate the mean, add a midpoint column $x$ and an $fx$ column, then use the same formula as before: estimated mean $=\frac{\sum fx}{\sum f}$. The word estimate is essential in your answer because the midpoint is only an assumption about where the values lie. Marks are commonly lost by using class boundaries instead of midpoints, so compute each midpoint carefully. You cannot give a single modal value from grouped data, so instead you state the modal class, the class interval with the highest frequency. If frequencies are 4, 11, 9 and 6 across four classes, the modal class is the second one. Be careful when class widths differ, but for a straightforward modal class you simply pick the tallest frequency. The class containing the median is found using cumulative frequency: locate the $\frac{n}{2}$th item for grouped data and read off which interval it falls in. You can also estimate the actual median by interpolation or by reading from a cumulative frequency graph, covered later. Remember that for grouped data every numerical answer beyond the modal class is an estimate.

Comparing Distributions and Statistical Diagrams

When a question asks you to compare two data sets, always compare two things: an average and a measure of spread, and write each comparison in context. A strong answer says, for example, that Class A has a higher median mark, so on average Class A scored better, and Class A has a smaller interquartile range, so its marks were more consistent. Two short sentences referring to the actual situation usually secure both comparison marks. Comparing only the averages, or quoting numbers without interpreting them, loses marks. Several diagrams display single data sets. A bar chart uses bars of equal width with gaps for categorical or discrete data, and the height shows the frequency. A pie chart shows proportions of a whole; each category gets a sector whose angle is $\frac{\text{frequency}}{\text{total}}\times 360^{\circ}$. To read a pie chart back, a sector angle of $90^{\circ}$ represents a quarter of the total. Always check your pie chart angles sum to $360^{\circ}$. A stem-and-leaf diagram keeps the original data while showing its shape: the stem is the leading digit (or digits) and each leaf is the final digit, written in order. A key such as 2 | 5 means 25 is essential and must be included. Because the values are preserved and ordered, you can read the median, mode and range straight from the diagram. A time series graph plots a measurement against time, with points joined by lines, and is used to spot a trend (a general upward or downward movement) and any seasonal pattern that repeats. Comment on the overall trend and on any regular ups and downs rather than describing every wobble.

Scatter Graphs, Correlation and Lines of Best Fit

A scatter graph plots paired data, one variable on each axis, to investigate whether two quantities are related. Correlation describes the relationship shown by the pattern of points. Positive correlation means that as one variable increases the other tends to increase too, so the points rise from left to right. Negative correlation means that as one increases the other tends to decrease, so the points fall from left to right. If the points show no pattern there is no correlation. You can also describe correlation as strong (points close to a line) or weak (points loosely scattered). A crucial exam point: correlation does not prove causation. Two variables can rise together because both are linked to a third factor, so never claim one causes the other from a scatter graph alone. A line of best fit is a straight line drawn through the middle of the points so that they are balanced on either side; it should follow the trend and need not pass through the origin. Draw it only when there is reasonable correlation. Use the line to make predictions: read across and up from a known value. A prediction made inside the range of the data is interpolation and is fairly reliable; a prediction outside the range is extrapolation and is unreliable because the pattern may not continue. An outlier on a scatter graph is a point that clearly does not fit the trend. Mention it if asked, and ignore it when positioning the line of best fit. Higher-tier questions may ask for the equation of the line in the form $y=mx+c$, found by reading the gradient and intercept from your drawn line.

Cumulative Frequency, Quartiles and Box Plots

Cumulative frequency is a running total of the frequencies, telling you how many data items are less than or equal to the upper boundary of each class. To draw a cumulative frequency graph you plot the cumulative frequency against the upper class boundary, then join the points with a smooth curve, giving the familiar S shape. Plotting against midpoints instead of upper boundaries is a frequent error. From the curve you can estimate key values for $n$ items. The median is read at the $\frac{n}{2}$ position on the cumulative frequency axis. The lower quartile (Q1) is at $\frac{n}{4}$ and the upper quartile (Q3) is at $\frac{3n}{4}$. Go across from these heights to the curve, then down to read the values. The quartiles divide the ordered data into four equal parts. The interquartile range (IQR) measures the spread of the middle half of the data: $\text{IQR}=Q_3-Q_1$. Because it ignores the top and bottom quarters, the IQR is not affected by outliers, which makes it a more reliable measure of spread than the range for skewed data. A smaller IQR means the data is more consistent. A box plot (box-and-whisker diagram) summarises five numbers: the minimum, lower quartile, median, upper quartile and maximum. The box spans from Q1 to Q3 with a line at the median, and the whiskers reach out to the minimum and maximum. Box plots are excellent for comparing two distributions at a glance: compare the medians for the averages and the IQRs (the box widths) for the spread, always writing the comparison in context.

Histograms and Frequency Density

A histogram looks like a bar chart but is used for continuous grouped data and, crucially, has no gaps between the bars because the scale is continuous. The big difference from a bar chart is that when the class widths are unequal you must plot frequency density on the vertical axis, not frequency. In a histogram it is the area of each bar, not its height, that represents the frequency. Frequency density is defined as frequency density $=\frac{\text{frequency}}{\text{class width}}$. So for a class running from 20 to 40 with a frequency of 30, the class width is 20 and the frequency density is $\frac{30}{20}=1.5$. Work out the class width as upper boundary minus lower boundary, and be careful with continuous boundaries (a class labelled 20 to 40 may have width 20). To read a frequency back from a histogram you reverse the formula: frequency $=$ frequency density $\times$ class width, which is exactly the area of the bar. This lets you find how many items fall in a part of a class. If a bar covers $30 \le x \lt 50$ and you want the count for $30 \le x \lt 40$, take the portion of the area over that range, assuming the data is spread evenly across the class. Histogram questions often give you some frequencies and some bar heights and ask you to complete the rest, then perhaps estimate the median or a total. Set out a table with columns for class width, frequency and frequency density so you can fill in whichever is missing. The key habit to build is: tall bar does not always mean most data; only the area tells you that.