Data analysis, statistics and the data booklet

Why numbers matter in SEHS

Every theme of the course rests on measurement: heart rate and VO₂ max in physiology, speed and force in biomechanics, questionnaire scores and accuracy in psychology and motor learning. These mathematical skills turn those measurements into evidence. They let you summarise a set of results, judge how variable and how certain they are, decide whether a difference or a relationship is real, and show it on a graph. They are examined in Paper 1B, can appear in any paper, and are the core of the data analysis in your scientific investigation.

Each skill is taught with a worked example using sport, exercise and health data (hypothetical unless a source is given) and the error students make most often. Your calculator handles the arithmetic; the marks are for choosing the right method, showing it and interpreting the answer.

Kinds of data and variables

Quantitative data are numbers (a time, a mass, a heart rate); qualitative data are descriptions (a participant reporting nausea, a coach's note that a jumper "stuttered" on the run-up). Quantitative variables are either continuous or discrete, and the difference decides how they are graphed and summarised.

Units, symbols and numbers

SI units, prefixes and conversions

SEHS uses SI units: the metre, kilogram, second, newton, joule and watt, with prefixes from the data booklet (kilo- 10³, milli- 10⁻³, micro- 10⁻⁶ and so on). Units are written with a space and negative indices: m s⁻¹, m s⁻², mL kg⁻¹ min⁻¹, beats min⁻¹. The data booklet also gives the conversions you need most: 1 hour = 60 minutes = 3600 seconds, 1 litre = 1 dm³, 1 joule = 0.239 calories, and g = 9.81 m s⁻².

Scientific notation and orders of magnitude

Very large and very small numbers are written in scientific notation: a number between 1 and 10 multiplied by a power of 10. An adult has about 5 × 10⁹ red blood cells in each millilitre of blood and about 5 L (5 × 10³ mL) of blood, so the total is about (5 × 10⁹) × (5 × 10³) = 2.5 × 10¹³ red blood cells. Multiplying numbers in scientific notation means multiplying the first parts and adding the powers.

An order of magnitude is a factor of 10. Comparing to the nearest order of magnitude is a quick way to check whether an answer is sensible. A sarcomere is about 2.5 µm (about 10⁻⁶ m) long and a muscle fibre is roughly 10–100 µm (10⁻⁵ to 10⁻⁴ m) across, so a fibre is one to two orders of magnitude wider than a sarcomere is long. A resting cardiac output of 5 L min⁻¹ and an elite maximal output of 35 L min⁻¹ are of the same order of magnitude.

Significant figures and decimal places

A calculated answer cannot be more precise than the data it came from. As a rule, give an answer to the same number of significant figures as the least precise value used. A distance of 5.00 m (3 significant figures) divided by a time of 0.49 s (2 significant figures) gives 10.204… m s⁻¹, which should be written 10 m s⁻¹ (2 significant figures), or 10.2 m s⁻¹ if the other values in the table are given to that precision. In a table of raw data, every value in a column is recorded to the same number of decimal places, matching the precision of the instrument.

Approximation, estimation and effects that can be ignored

Scientists often simplify a calculation by ignoring an effect that is small compared with the others. For a shot put, drag is tiny compared with the shot's weight, so the flight is treated as a projectile under gravity alone; for a badminton shuttle, drag is large and cannot be ignored. Estimating before calculating ("about 10 m s⁻¹ for a sprinter") catches slips such as a misplaced decimal point.

Fractions, ratios, percentages and reciprocals

Ratios compare two quantities: a work-to-rest ratio of 1:3 means 30 seconds of rest after a 10-second sprint. Percentages express a part of a whole out of 100: 18 successful serves out of 24 is 18 ÷ 24 × 100 = 75%. A reciprocal is 1 divided by a number, and it turns a quantity "per unit" into its inverse. Running pace and speed are reciprocals: a pace of 4.0 min km⁻¹ means 4.0 minutes for each kilometre, so the speed is 1 ÷ 4.0 = 0.25 km min⁻¹, which is 15 km h⁻¹ or 4.2 m s⁻¹.

Measures of central tendency

A measure of central tendency gives one typical value for a data set. The three in the course are the mean, the median and the mode.

• The mean uses every value, so it is the most informative when the data are roughly symmetrical, but one extreme value can pull it a long way.

• The median is the middle value when the data are in order (or the mean of the two middle values when there is an even number). It is not affected by extreme values, so it suits skewed data and data with outliers.

• The mode is the most frequent value. It is the only average for categorical data (the most common injury type) and can be misleading for small sets of continuous data.

Measures of spread

Two teams can have the same mean jump height but very different players. Measures of spread (dispersion) describe how far the values are scattered. The course uses five: the range, the standard deviation, the coefficient of variation, the standard error and the interquartile range.

Range

The range is the largest value minus the smallest. It is quick to calculate but depends on only two values, so a single outlier changes it completely. It is used for error bars in some small data sets, and to describe the spread of repeated trials ("the three trials ranged from 41 to 46 kg").

Standard deviation

For a sample of n values with mean $\bar{x}$, the standard deviation s is:

In words: find how far each value is from the mean, square those deviations so they are all positive, add them, divide by n − 1, and take the square root. Dividing by n − 1 instead of n corrects for the fact that a sample tends to underestimate the spread of the whole population. In practice you use a calculator (the sx or σn−1 key) or a spreadsheet (STDEV.S); the formula is not in the data booklet, but you must know what SD means and how to use it.

When data follow a normal distribution, the SD gives a precise picture of where most values lie. The data booklet shows the normal distribution with its three bands.

If the VO₂ max of a large group of young adults is normally distributed with a mean of 45 mL kg⁻¹ min⁻¹ and an SD of 6 mL kg⁻¹ min⁻¹ (hypothetical values), about 68% of them lie between 39 and 51 mL kg⁻¹ min⁻¹ and about 95% between 33 and 57 mL kg⁻¹ min⁻¹. A person at 60 mL kg⁻¹ min⁻¹ is 2.5 SDs above the mean, which puts them among the top 1% or so of the group. Many fitness test norm tables are built this way.

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Coefficient of variation

The SD is in the units of the data, so it cannot compare the spread of two different measures. An SD of 0.08 s in 20 m sprint times and an SD of 5.1 cm in jump heights cannot be compared directly. The coefficient of variation solves this by expressing the SD as a percentage of the mean. The data booklet gives it as:

The coefficient of variation is also the usual way to report the reliability of a test. If an athlete repeats the same test several times, a low V between trials shows that the test gives consistent results. There is no single agreed cut-off for good reliability: the acceptable value depends on how small a change the test needs to detect, so a test meant to detect a 2% improvement needs a lower V between trials than one meant to detect a 10% improvement.

Standard error

The SD describes the spread of individual values. The standard error describes the uncertainty in the mean itself: if the study were repeated with a new sample, how much would the mean change? Because SE = SD ÷ √n, the standard error falls as the sample gets bigger, while the SD does not. A sample of 16 participants with an SD of 6.2 beats min⁻¹ has an SE of 6.2 ÷ √16 = 6.2 ÷ 4 = 1.55 beats min⁻¹. The formula is not in the data booklet.

The figure shows why the caption of a graph matters. With SD bars, the caffeine and placebo groups overlap, which shows that many individual cyclists in the two groups had similar times. With SE bars, the means are clearly separated, which suggests that the group means are precise enough to differ. Both versions are correct: SD bars show how different the people were, and SE bars show how certain the means are.

Interquartile range

The quartiles divide ordered data into four equal parts, as the data booklet's diagram shows: Q1 has 25% of the values below it, the median (Q2) 50% and Q3 75%. The interquartile range is Q3 − Q1. It is the measure of spread that goes with the median, and it is shown on a box plot.

Percentage change, difference and error

Percentages appear in almost every 1B question. Three different calculations share the word, and each divides by a different value.

Percentage change

Percentage change measures how much a value has risen or fallen relative to where it started, so it is the right calculation for any before-and-after comparison.

Percentage difference

Percentage difference compares two values when neither is the "original", for example two methods of measuring the same thing, or two athletes. It divides the difference by the mean of the two values:

If a 100 m run is timed at 11.42 s by hand and at 11.66 s by electronic timing (hypothetical values), the percentage difference is 0.24 ÷ 11.54 × 100 = 2.1%. Hand timing usually gives faster times, because the timekeeper reacts to the start signal late but anticipates the finish.

Percentage error

Uncertainty in measurements

No measurement is exact. Every value recorded in an investigation has an uncertainty, written as a range: 72.4 ± 0.1 kg. The uncertainty of a single reading depends on the instrument. For a digital instrument it is usually the smallest reading it can show (± 0.1 kg for scales that read to 0.1 kg). For an analogue scale such as a tape measure, it is usually half the smallest division (± 0.5 mm on a tape marked in millimetres), unless the method makes it larger. Human timing adds reaction time, so a hand-held stopwatch reading to 0.01 s is often given an uncertainty of about ± 0.2 s.

When a measurement is repeated, the spread of the repeats also gives an estimate of the uncertainty: half the range of the repeats is a common choice. Uncertainties are usually rounded to one significant figure, and the value is given to the same decimal place as its uncertainty: 6.6 ± 0.1 m s⁻¹, not 6.637 ± 0.1 m s⁻¹.

Percentage uncertainty compares the uncertainty with the size of the measurement. A hand-timed 12.4 ± 0.2 s has a percentage uncertainty of 0.2 ÷ 12.4 × 100 = 1.6%. The same ± 0.2 s on a 3.1 s sprint would be 6.5%, which is why hand timing is unsuitable for short sprints.

Propagating uncertainties

When measured values are combined in a calculation, their uncertainties combine too. The data booklet gives three rules:

The second rule applies to every equation in the data booklet that multiplies or divides, such as speed, acceleration, force, momentum and power. Fractional uncertainties are added because a small error in either the top or the bottom of a fraction changes the answer by a similar fraction.

The third rule is needed when a measured value is squared or raised to another power. Body mass index (BMI) divides mass by height squared, so it combines the second and third rules.

Proportionality and correlation

Direct and inverse proportionality

Two variables are directly proportional when doubling one doubles the other; the graph is a straight line through the origin. Newton's second law, $F=ma$, gives an example: for a fixed mass, the force needed is directly proportional to the acceleration. Two variables are inversely proportional when doubling one halves the other. The time to cover a fixed distance is inversely proportional to speed, because $t=\frac{d}{s}$.

Many relationships in SEHS are neither: they rise and then level off (heart rate against time at a steady workload), or have a peak (performance against arousal). Proportional language should be kept for relationships that really are proportional.

Correlation and the correlation coefficient

A scatter graph shows whether two continuous variables are correlated. The correlation coefficient r measures how closely the points follow a straight line, from +1 (a perfect positive line) to −1 (a perfect negative line); 0 means no linear relationship.

A sample can show a correlation by chance, especially a small one, so r is compared with a critical value that depends on the number of pairs of data. The degrees of freedom are n − 2. If the size of r (ignoring its sign) is greater than the critical value at p = 0.05, the correlation is significant and the null hypothesis of no relationship is rejected. Critical values are given in the question when you need them; the table shows typical values for Pearson's r.

The coefficient of determination, R²

A spreadsheet adds R² to a trend line to show how well the line fits the data. R² = 0.80 means that 80% of the variation in the y-values is explained by the trend line, and 20% is due to other factors or random variation.

In a hypothetical study of 12 long jumpers with r = 0.89 between approach speed and jump distance, so R² = 0.89² = 0.79: about four-fifths of the differences in jump distance between the athletes go with differences in approach speed. Comparing R² for a straight line and a curve helps choose the better model, but a curve with more bends always fits slightly better, so the shape must also make physiological or physical sense.

The t-test and statistical significance

A difference between two means could be a real effect of the independent variable or a chance result of which participants happened to be in the sample. A test of statistical significance decides between these. The test begins from the null hypothesis that there is no difference between the populations the samples came from.

The t-test produces a value of t: the larger t is, the less likely the difference is to be due to chance. The value of t is compared with a critical value for the degrees of freedom at p = 0.05. If t (ignoring its sign) is greater than the critical value, p < 0.05 and the difference is significant: the null hypothesis is rejected. A spreadsheet (for example the T.TEST function) gives the p-value directly.

For two groups of equal size, the unpaired t is the difference in means divided by its standard error:

A paired t-test uses the difference for each participant. t is the mean of the differences divided by its standard error, and the degrees of freedom are the number of pairs minus 1:

Graph skills

Choosing the right graph

Plotting a graph

A plotted graph needs the independent variable on the x-axis and the dependent variable on the y-axis, each with a label and a unit ("Heart rate (beats min⁻¹)"). Scales go up in equal steps that are easy to read (1, 2, 5 or 10 per square) and use most of the grid. An axis does not have to start at zero, but a break symbol shows when it does not. Points are plotted as small crosses or dots; if more than one data set shares the axes, each has its own symbol and a key, and the title says what the graph shows.

Sketch graphs

A sketch shows the general shape of a relationship on labelled axes without a scale: the inverted U, oxygen deficit and EPOC, heart rate before, during and after exercise, the path of a projectile. It still needs labelled axes with the variables (units are not needed without values), the correct shape, and the key features marked: a peak, a plateau, the point where exercise starts. Sketching is a drawing skill; plotting is a measuring skill.

Lines of best fit, gradients, intercepts and areas

A line of best fit shows the trend in scattered data. For a straight line, it passes through the middle of the points with about as many above as below; it does not have to pass through the origin or join the first and last points. A curve of best fit is drawn smoothly when the trend is clearly curved. The gradient of the line gives the rate of change of y with x, in the units of y divided by the units of x, and the intercept is the value of y where the line crosses the y-axis (when x = 0).

On a velocity–time graph the gradient is the acceleration and the area under the line is the displacement. The same ideas apply to other graphs: the gradient of a heart rate–time graph is the rate at which heart rate rises, and the area between an oxygen uptake curve and the steady-state demand is the oxygen deficit.

Plotting error bars

Error bars are drawn as a vertical line through each mean, from the mean minus the chosen measure to the mean plus it, with a short cap at each end. A mean of 147 beats min⁻¹ with an SD of 10 beats min⁻¹ has a bar from 137 to 157. The key or caption must say whether the bars show ±1 SD, ±1 SE, the range or the uncertainty of the instrument. On a scatter graph, horizontal and vertical bars can show the uncertainty in both variables; a line of best fit that passes through every bar is consistent with the data.

Interpolation and extrapolation

Interpolation reads a value from the line of best fit within the range of the data; it is usually reliable. Extrapolation extends the line beyond the data to predict a value; it assumes the trend continues, which may not be true. Submaximal fitness tests depend on extrapolation: heart rate is measured at several low work rates and the line is extended to the age-predicted maximum heart rate to estimate the maximal work rate or VO₂ max.

Graphs also have features worth naming in a description: a maximum or minimum (peak heart rate, the lowest blood glucose), a change in gradient (the point where blood lactate starts to rise sharply as intensity increases), a plateau (steady state) and an intercept (the resting value at time zero).

The data booklet

Every candidate has a clean copy of the SEHS data booklet in every paper. It is a reference, so it saves you memorising some content, but only if you know what is in it and where. The table lists its seventeen sections.

Some tools you need are not in the booklet: the formulas for the mean, standard deviation, standard error, percentage change, percentage difference and percentage error; how to carry out a t-test; and the critical value tables for t and r, which are given in the question when they are needed. These must be learned.

Checklist: data analysis skills

• To do:

  I can calculate a mean, median, mode and range, and choose the right average for the data.

• To do:

  I can calculate and interpret a standard deviation, and use the 68–95–99.7% rule.

• To do:

  I can calculate the coefficient of variation with the data booklet equation and use it to compare spread.

• To do:

  I can calculate a standard error and explain how SE bars differ from SD bars.

• To do:

  I can find the quartiles and interquartile range and draw a box plot.

• To do:

  I can calculate percentage change, difference and error, and say which one a question needs.

• To do:

  I can record uncertainties and propagate them with the data booklet rules.

• To do:

  I can compare r with a critical value, interpret R² and explain why correlation is not causation.

• To do:

  I can choose a paired or unpaired t-test, calculate t and interpret it at p = 0.05.

• To do:

  I can plot and sketch graphs, draw lines of best fit, find gradients and areas, and interpolate and extrapolate with care.