Topic 9: Statistics

Statistics is the branch of mathematics concerned with collecting, organising, displaying and interpreting numerical information. In the IGCSE 0580 course you are expected both to calculate measures (averages and spread) and to represent data using a wide range of diagrams. The skills divide naturally into three families: averages and spread (mean, median, mode, range, quartiles, interquartile range), data displays (bar charts, pie charts, pictograms, stem-and-leaf, scatter diagrams, box plots, histograms), and the techniques for grouped data (estimated mean, modal class, frequency density). This article walks through each in turn with worked formulas so you can both compute confidently and read information back out of any chart the exam puts in front of you.

Collecting and Classifying Data

Before any calculation you must understand what kind of data you have. Data is either qualitative (descriptive, such as eye colour) or quantitative (numerical). Quantitative data is further split into discrete data, which can only take separate fixed values (the number of pets in a house must be a whole number), and continuous data, which can take any value in a range (height, mass, time). This distinction matters because continuous data is usually grouped into class intervals and drawn as a histogram, while discrete data is often shown in a bar chart. When classifying continuous data we use intervals written with inequalities, for example $10 \\lt h \\leq 20$, where every value belongs to exactly one class. Recording raw values in a tally chart and then a frequency table is the first step of organising data: the frequency is simply how many items fall in each category or class. Always check that the frequencies add up to the total number of items, $\\sum f = n$, as a quick validity check before you go further.

Mean, Median, Mode and Range from a List

For a simple list of values there are three averages and one common measure of spread. The mean is the sum of the values divided by how many there are: $\\bar{x} = \\frac{\\sum x}{n}$. The median is the middle value once the data is arranged in order; for $n$ values its position is $\\frac{n+1}{2}$, so with an odd $n$ you read off a single middle value, and with an even $n$ you average the two central values. The mode is the value that occurs most often, and a data set may have one mode, several modes, or no mode at all. The range measures spread and is the largest value minus the smallest value, $\\text{range} = \\text{max} - \\text{min}$. For example, for the list 4, 7, 7, 2, 10 the ordered list is 2, 4, 7, 7, 10, giving median 7, mode 7, range $10 - 2 = 8$, and mean $\\frac{30}{5} = 6$. Remember the median needs ordered data first, a mistake that costs easy marks.

Averages from a Frequency Table

When data is summarised in a frequency table you must weight each value by its frequency rather than treating the table as a short list. The mean becomes $\\bar{x} = \\frac{\\sum fx}{\\sum f}$, where you create an $fx$ column by multiplying each value $x$ by its frequency $f$, total that column, and divide by the total frequency $\\sum f$. The mode is the value with the highest frequency, read directly from the table. The median is the value at position $\\frac{n+1}{2}$ where $n = \\sum f$; to find it you work along the frequencies, accumulating them until you reach that position. For instance, if 30 students scored marks with frequencies adding to 30, the median lies at position $\\frac{30+1}{2} = 15.5$, so you average the 15th and 16th values found by counting through the cumulative frequencies. A frequent error is dividing $\\sum fx$ by the number of rows instead of by $\\sum f$, so always divide by the total frequency.

Grouped Data: Estimated Mean and Modal Class

With grouped continuous data the individual values are lost inside each class, so you can only estimate the mean. The method is to use the midpoint of each class as a representative value: for a class $10 \\lt x \\leq 20$ the midpoint is $\\frac{10+20}{2} = 15$. You then apply the same formula as a frequency table using midpoints, $\\bar{x} \\approx \\frac{\\sum fx}{\\sum f}$, where $x$ now means the midpoint of each class. The result is an estimate because each value is assumed to sit at the centre of its class. For grouped data the mode is replaced by the modal class, which is simply the class interval with the highest frequency, not a single number. You cannot find an exact median for grouped data either; instead you identify the median class as the one containing the $\\frac{n}{2}$-th value, or you read an estimate from a cumulative frequency curve as described later. Always state your answer as an estimate and label the modal class as an interval.

Charts: Bar Charts, Pie Charts, Pictograms and Stem-and-Leaf

Several diagrams display data visually. A bar chart uses bars of equal width whose heights show frequency, with gaps between bars for discrete or categorical data. A pictogram uses a symbol to represent a fixed number of items, so you must read the key carefully and may need to interpret part-symbols. A pie chart represents the whole data set as a circle of $360$ degrees, with each category given a sector whose angle is proportional to its frequency: $\\text{angle} = \\frac{f}{\\sum f} \\times 360$. To read a pie chart in reverse, a sector of angle $a$ represents a fraction $\\frac{a}{360}$ of the total. A stem-and-leaf diagram keeps the actual data values while ordering them: the stem holds the leading digits and each leaf a final digit, so 34 and 37 appear as a stem of 3 with leaves 4 and 7. An ordered stem-and-leaf diagram is excellent for quickly finding the median, mode and range because the raw values remain visible, and it must always include a key such as 3 | 4 means 34.

Scatter Diagrams, Correlation and Line of Best Fit

A scatter diagram plots paired data, one variable on each axis, to investigate whether a relationship exists between them. The pattern of points describes the correlation. Positive correlation means that as one variable increases the other tends to increase, so the points slope upward; negative correlation means one increases as the other decreases, sloping downward; and if the points show no pattern there is no (or zero) correlation. The strength is judged by how closely the points cluster around a straight line: tightly clustered points show strong correlation, scattered points show weak correlation. When the data shows correlation you can draw a line of best fit, a straight line passing through the general trend with roughly equal numbers of points on each side, ideally passing close to the mean point $(\\bar{x}, \\bar{y})$. This line lets you estimate a missing value by reading across and up from the line. Be cautious: estimating outside the range of the data (extrapolation) is unreliable, and correlation does not by itself prove that one variable causes the other.

Cumulative Frequency, Median, Quartiles and Box Plots

Cumulative frequency is a running total of frequencies, telling you how many data items are less than or equal to the upper boundary of each class. Plotting cumulative frequency against the upper class boundary and joining the points with a smooth curve gives a cumulative frequency curve, an S-shaped graph that lets you estimate key values. With a total of $n$ items, the median is read at the $\\frac{n}{2}$ position on the vertical axis, the lower quartile $Q_1$ at $\\frac{n}{4}$, and the upper quartile $Q_3$ at $\\frac{3n}{4}$; in each case you go across to the curve and down to the horizontal axis to read the value. The interquartile range measures the spread of the middle half of the data and is $\\text{IQR} = Q_3 - Q_1$, a measure that ignores extreme values and so is more robust than the range. These five numbers (minimum, $Q_1$, median, $Q_3$, maximum) form the five-number summary used to draw a box plot, where a box spans $Q_1$ to $Q_3$ with a line at the median and whiskers extend to the minimum and maximum. Box plots are ideal for comparing two data sets at a glance by their median and IQR.

Histograms with Frequency Density

A histogram looks like a bar chart but is used for continuous grouped data and, crucially, may have unequal class widths. To make the comparison fair the area of each bar, not its height, represents the frequency. The vertical axis therefore shows frequency density, defined as $\\text{frequency density} = \\frac{\\text{frequency}}{\\text{class width}}$. To draw a histogram you calculate the class width and the frequency density for each class, then draw bars with no gaps because the data is continuous. To read information back out, you reverse the formula: $\\text{frequency} = \\text{frequency density} \\times \\text{class width}$, which equals the area of the bar. A common exam task gives some frequencies and some frequency densities and asks you to complete the table, so be confident moving in both directions. The total area of all bars equals the total frequency $\\sum f$, a useful check. Never plot raw frequency on the vertical axis when class widths differ, as this would distort the picture; frequency density is what keeps a histogram honest.