4.3 Measures of central tendency and dispersion

Measures of central tendency

A measure of central tendency gives a single number for a typical value of a data set. There are three in common use (the mean, the median and the mode), and each defines "typical" in a different way. Each suits some kinds of data and contexts better than others.

The mean

The mean is the equal share of the total. If five friends have $3, $7, $8, $10 and $12, together they have $40, and if they pooled the money and shared it equally each would have $40 \div 5 = 8$ dollars. The mean is the value every data item would take if the total were spread evenly across all of them.

For $n$ values $x_1, x_2, \ldots, x_n$, the definition becomes a formula using sigma notation, where $\sum$ means "add up".

The mean has a balancing property that explains why it is called the centre. The distances of the values from the mean, counted as negative for values less than the mean and positive for values greater than it, always add up to zero. For 3, 7, 8, 10, 12 these distances are $-5, -1, 0, 2, 4$, and their sum is 0. If the values were weights on a see-saw, the mean is the point where it balances. This property returns in the derivation of the standard deviation.

The mean from a frequency table

Data with many repeated values are usually given in a frequency table, in which each value $x_i$ appears with its frequency $f_i$. Adding a value that occurs $f_i$ times is the same as multiplying it by $f_i$, so the total of all the data is $\sum f_i x_i$ and the number of data items is $\sum f_i$. This gives the formula for the mean of a frequency table.

Here $k$ is the number of different values in the table (the number of rows), while $n$ is the number of data items. Dividing by $k$ instead of $n$ is one of the most common errors in statistics.

A survey records the number of children in each of 60 households.

The median

The median is the middle value when the data are arranged in order, so half the data are at or less than it and half are at or greater than it. For $n$ ordered values it is the value in position $\frac{n+1}{2}$. When $n$ is odd this is a single value; when $n$ is even it falls halfway between two values, and the median is the mean of those two.

The formula gives the position of the median, not its value. For 11 values the median is the 6th value; for 60 values the position is 30.5, so the median is halfway between the 30th and 31st values. In a frequency table, a running total of the frequencies (the cumulative frequency) locates any position quickly. For the 60 households, the running totals are 8, 23, 45, … ; the 30th and 31st values both lie in the group whose running total first passes 30, which is $x = 2$, so the median is 2 children.

The mode

The mode is the value that occurs most often. It is the only measure of central tendency that also works for qualitative data: the modal colour of cars sold or the modal choice of transport to school makes sense, while a mean colour does not.

A data set with two values tied for the highest frequency is called bimodal. Bimodal data often signal two different groups mixed together, such as the heights of a class containing both 11-year-olds and 17-year-olds. In such cases the mean and median can fall in the gap between the two groups, where few values actually lie.

Finding unknown values from a given mean

Because the mean formula connects the total, the number of items and the mean, any one of them can be found from the other two. Rearranging $\bar{x} = \frac{\sum f_i x_i}{n}$ gives $\sum f_i x_i = n\bar{x}$: the total equals the mean multiplied by the number of items. When a frequency or a data value is unknown, substituting into the formula produces a linear equation; with two unknowns, a second fact (such as the total frequency) gives a second equation, and the pair is solved as a system of linear equations (subtopic 1.8, Equations and systems using technology).

The number of goals scored by a netball club's junior team in each match of a season is recorded in a table with one unknown frequency, $k$.

With two unknowns, two equations are needed. The IB grades of 50 students are shown with two frequencies missing.

Choosing a measure of central tendency

The three measures behave differently when the data contain extreme values. The mean uses the size of every value, so a single very large or very small value moves it. The median uses only the order of the values, so it changes little however extreme the largest or smallest value becomes. The mode depends only on which value occurs most often.

A famous basketball game shows the difference clearly. On 2 March 1962, Wilt Chamberlain scored 100 points for the Philadelphia Warriors in a 169–147 win over the New York Knicks, still the highest score by one player in a single NBA game. The points scored by the nine Philadelphia players who took part are shown in the table.

Source: official box score, Philadelphia Warriors v New York Knicks, 2 March 1962, as reproduced by NBA.com.

The mean is $\frac{169}{9} = 18.8$ points, but seven of the nine players scored fewer than that, and the only player who scored many more is Chamberlain. The median, the 5th of the nine ordered values (0, 0, 0, 9, 11, 16, 16, 17, 100), is 11 points. The dot plot of the nine scores shows the mean pulled far to the right of the cluster by one value.

Removing Chamberlain's score shows how much each measure depends on one value. The remaining eight scores have mean $\frac{69}{8} = 8.63$ points (a fall of more than half) and median $\frac{9 + 11}{2} = 10$ points (a fall of 1). A measure that hardly changes when extreme values are added, removed or altered is called robust.

Chamberlain's 100 is an outlier by the 1.5 × IQR rule of subtopic 4.1: for these nine scores $Q_1 = 0$ and $Q_3 = 16.5$, so any value greater than $16.5 + 1.5 \times 16.5 = 41.25$ is an outlier. It is a genuine value rather than a recording error, so it stays in the data. The right response to a genuine outlier is to choose a summary that is not distorted by it, or to report both measures and explain the difference.

How far apart the mean and median are depends on the shape of the distribution. In a symmetric distribution the mean and median are equal or nearly equal. In a positively skewed distribution, with a long tail of large values, the mean is greater than the median; in a negatively skewed distribution the mean is less than the median. Incomes, house prices and waiting times are typically positively skewed, which is why their medians are usually quoted.

Grouped data: the estimated mean and the modal class

Large data sets of continuous data (times, masses, lengths) are usually presented as grouped data, with values collected into class intervals such as $20 \le t < 30$. Grouping makes the data easier to read but loses the individual values: the table shows that 23 people took between 20 and 30 minutes, but not whether they took 21 minutes or 29. The mean of grouped data therefore cannot be calculated exactly; it can only be estimated.

Mid-interval values

The estimate assumes that every value in a class sits at the middle of the class. Some values in the class are less than the middle and some greater, so the errors tend to cancel out when the class's values are spread fairly evenly.

For continuous data the mid-interval value is the mean of the two class boundaries. For discrete data grouped as 11–20, 21–30, …, the boundaries lie halfway between the classes (10.5 and 20.5), so the mid-interval value of 11–20 is $\frac{10.5 + 20.5}{2} = 15.5$, which equals $\frac{11 + 20}{2}$. For a class such as $100 < x \le 150$, the mid-interval value is 125.

The estimated mean

Replacing each value by the mid-interval value of its class turns the grouped table into an ordinary frequency table, and the mean formula applies with $x_i$ as the mid-interval values.

The result is an estimate because the true values in each class are unknown. It is close to the true mean when values are spread evenly within classes and the classes are narrow; it can be poor when classes are wide and values crowd towards one end of them.

The commuting times, $t$ minutes, of 80 employees at an office are recorded in a grouped table.

The modal class

With grouped data the single most common value is not known, but the class containing the most values is.

The equal-width condition matters. A class twice as wide as its neighbours collects roughly twice as many values for the same density of data, so with unequal widths the class with the highest frequency is not necessarily where the data are most concentrated. Published income tables, whose brackets often have different widths, are a common example of this trap.

The median of grouped data, like the quartiles, is estimated from a cumulative frequency graph (subtopic 4.2, Presenting data). The table identifies the class that contains the median: for 80 commuters the median is about the 40th value, and the running totals 6, 20, 43, … show it lies in $20 \le t < 30$.

Measures of dispersion

Two data sets can share a centre and still look completely different. A measure of dispersion puts a number on the spread.

Range and interquartile range

Two measures of dispersion come from subtopic 4.2. The range is the largest value minus the smallest; it is quick but depends on the two most extreme values only, so a single outlier changes it completely. The interquartile range (IQR) is the spread of the middle half of the data, $\mathrm{IQR} = Q_3 - Q_1$, where the lower quartile $Q_1$ has a quarter of the data at or less than it and the upper quartile $Q_3$ has three quarters at or less than it. Because it ignores the top and bottom quarters, the IQR is robust, and it is the natural partner of the median. Neither measure uses every value; the standard deviation uses them all, and it is the measure of spread that goes with the mean.

Deviations from the mean

To measure how far values sit from the mean, the natural starting point is each value's distance from the mean.

Averaging the deviations fails as a measure of spread, because the balancing property of the mean makes them sum to zero for every data set. The negative and positive deviations cancel. Squaring each deviation removes the signs (every square is positive or zero), and averaging the squares gives a measure that is large when values are far from the mean and small when they are close to it. That average of squared deviations is the variance.

The variance has one drawback: its units are the square of the data's units. Heights in centimetres have a variance in square centimetres, which cannot be compared with the heights themselves. Taking the square root returns to the original units.

The relationship works in both directions: the variance is the square of the standard deviation, and the standard deviation is the square root of the variance. Because squaring gives the largest deviations the greatest weight, the standard deviation is strongly affected by outliers. It is not robust: for the nine Philadelphia scores $\sigma = 29.5$ points, but without Chamberlain's 100 it falls to $\sigma = 7.14$ points.

Calculating the standard deviation by hand

Standard deviation and variance are calculated with technology, but one calculation by hand shows what the GDC does. The table method keeps the working organised.

Other forms of the variance formula

Expanding the squared deviations gives an equivalent formula that some textbooks and calculators use: the mean of the squares minus the square of the mean.

For 4, 7, 8, 10, 11 the sum of the squares is $16 + 49 + 64 + 100 + 121 = 350$, so $\sigma^2 = \frac{350}{5} - 8^2 = 70 - 64 = 6$, as before. The two forms always agree; the second needs only the totals $\sum x$ and $\sum x^2$, which is why GDCs display them.

A different version of the formula divides by $n - 1$ instead of $n$. It is used when the data are a sample and the aim is to estimate the spread of the whole population from which the sample came. GDCs show both results side by side: $\sigma_x$ (dividing by $n$) and $s_x$ or $\mathrm{S}x$ (dividing by $n - 1$). Unless a question says otherwise, the data set is treated as the population, so the value to use is $\sigma_x$.

Standard deviation and variance with the GDC

The GDC's one-variable statistics function calculates the mean, the standard deviation, the quartiles and more in one step. The method is the same for raw data and for frequency tables.

1. Enter the data values in one list (for grouped data, the mid-interval values).

2. If the data are in a frequency table, enter the frequencies in a second list.

3. Run one-variable statistics, choosing the data list and, for a frequency table, the frequency list. On a TI-84 Plus CE: STAT → CALC → 1-Var Stats, then set List and FreqList. On a Casio fx-CG50: MENU → Statistics, then CALC → SET to choose the list and frequency, then 1-VAR.

4. Read $\bar{x}$, $\sigma_x$, $n$ and the five-number summary (minimum, $Q_1$, median, $Q_3$, maximum).

5. For the variance, square $\sigma_x$ using its unrounded value.

The output contains two standard deviations, and choosing the wrong one is the most frequent GDC error in descriptive statistics.

The standard deviation of grouped data is estimated in the same way as the mean, by entering the mid-interval values with the frequencies. For the 80 commuters, the GDC gives $\sigma_x = 13.87\ldots$, so the estimated standard deviation is 13.9 minutes. Like the estimated mean, it is approximate, because every time in a class has been replaced by the class's mid-interval value.

Interpreting the standard deviation

A standard deviation is meaningful only beside the mean and in the units of the data. A larger standard deviation means the values are more spread out from the mean, so they are less consistent; a smaller one means they cluster closely around the mean. A standard deviation of 0 means every value is the same.

The following data are hypothetical. A food company fills bags labelled 500 g on two machines and weighs ten bags from each.

Both machines have mean 500 g, so the means alone would suggest they perform equally well. The standard deviations are 1.10 g for machine A and 6.39 g for machine B, and the dot plots of the two samples show what those numbers mean.

Machine B's bags are much less consistent: some customers receive 489 g and others 510 g, so the company gives away product in some bags and under-fills others. Machine A is the better machine, even though both have the same mean. The standard deviation also gives a natural yardstick. For machine B, the interval from one standard deviation below the mean to one standard deviation above it, $493.6 \le m \le 506.4$, contains 6 of the 10 bags. For many roughly symmetric, bell-shaped data sets about two-thirds of the values lie within one standard deviation of the mean, an idea developed with the normal distribution in subtopic 4.9.

Quartiles of discrete data

For raw or tabulated discrete data, the quartiles come from the GDC's one-variable statistics, along with the median. The idea is the same as for the median: $Q_1$ splits off the lowest quarter of the ordered data and $Q_3$ the highest quarter. For small data sets there is no single agreed rule for exactly where those splits fall, because a quarter of 10 values is 2.5 values.

The method most GDCs use

Most GDCs use a "median of halves" rule.

1. Order the data and find the median.

2. Split the data into a lower half and an upper half. When $n$ is odd, the median itself is left out of both halves.

3. $Q_1$ is the median of the lower half and $Q_3$ is the median of the upper half.

For the 11 hockey scores 0, 1, 1, 2, 2, 2, 3, 3, 4, 5, 6, the median (the 6th value, 2) is left out. The lower half 0, 1, 1, 2, 2 has median 1 and the upper half 3, 3, 4, 5, 6 has median 4. So $Q_1 = 1$, $Q_3 = 4$ and $\mathrm{IQR} = 4 - 1 = 3$ goals.

Different methods give different quartiles

Spreadsheets and statistics software often use a different rule. A common one (the spreadsheet function QUARTILE.INC, and the default in many programming libraries) places $Q_1$ at position $1 + 0.25(n - 1)$ in the ordered data and $Q_3$ at $1 + 0.75(n - 1)$, interpolating between neighbouring values when the position is not a whole number.

The same median-of-halves rule applies to a frequency table, with the positions located from the running totals. For the 60 households, the lower half is the 1st to 30th values and its median is halfway between the 15th and 16th values, both 1, so $Q_1 = 1$. The upper half is the 31st to 60th values, whose median is halfway between the 45th value (2, since the running total reaches 45 at $x = 2$) and the 46th value (3), so $Q_3 = 2.5$. The IQR is $2.5 - 1 = 1.5$ children. A GDC with the frequency list set gives the same values.

Effect of constant changes on the data

Data are often transformed by the same rule applied to every value: converting units (°C to °F, miles to kilometres, euros to dollars), adding a bonus to every test mark, or scaling raw marks to a standard scale. The summary measures of the new data can be found directly from those of the old data, without recalculating anything.

Adding or subtracting a constant

Adding the same constant $b$ to every value moves the whole data set along the number line by $b$ without changing its shape. Every measure of position moves by $b$: the mean, median, mode and quartiles all increase by $b$. The distances between values do not change, so no measure of spread changes: the range, IQR, standard deviation and variance stay the same.

The algebra confirms it. If $y_i = x_i + b$, the new total is $\sum f_i x_i + nb$, so the new mean is the old mean plus $b$, and each new deviation equals the old one.

Since the deviations are unchanged, so are the variance and standard deviation, which are built from them.

Multiplying by a constant

Multiplying every value by a positive constant $a$ stretches the number line by a factor $a$. Every measure of position is multiplied by $a$, and so is every distance between values, so the range, IQR and standard deviation are also multiplied by $a$. The variance, which is built from squared distances, is multiplied by $a^2$.

The dot plots of 2, 4, 5, 7, 12, after adding 3 to every value and after doubling every value, show the two effects.

The general rule

Both changes together give the rule for any linear change $y = ax + b$ with $a > 0$.

The order of operations matters for the measures of position: in $y = 2x + 20$, the mean is doubled first and then 20 is added. For the measures of spread, the added constant never appears. If $a$ is negative (for example $y = 100 - x$, which reverses the order of the data), the spread is multiplied by $|a|$, since a standard deviation can never be negative.

Unit conversion: Hong Kong temperatures in °C and °F

The Hong Kong Observatory publishes the average temperature of each month at its headquarters over the 30 years 1991 to 2020.

Source: Hong Kong Observatory, Monthly Meteorological Normals for Hong Kong (1991–2020).

The twelve monthly means are averaged with equal weight here, although the months have different lengths. Weighting each month by its number of days gives 23.55 °C, only 0.03 °C higher, and the constant-change rules apply to either version in exactly the same way.

Case study: US household income in 2024

The US Census Bureau's Current Population Survey estimated the income of the country's 134.8 million households in 2024. It reported a median household income of $83,730 and a mean household income of $121,000. The mean is about 45% larger than the median. The percentage distribution of households by income bracket shows why.

Source: US Census Bureau, Income in the United States: 2024 (Current Population Survey, Annual Social and Economic Supplement), Table A-2, published 2025; mean income of the top bracket from Table HINC-06 (2025).

The classes have unequal widths, from $10,000 to $50,000, and the last class has no upper limit. To compare classes fairly, the histogram plots the percentage of households per $10,000 of income, so that the area of each bar matches its percentage.

Shape and choice of measure. The distribution is strongly positively skewed: incomes cannot fall much below zero, but they can be many times the median. The households in the open top class include some with incomes in the millions, and each of them raises the mean but changes the median no more than any other household with an income above the median would. The median is therefore the better description of a typical household, and it is the figure usually quoted for a typical household. The mean is still useful: multiplied by the number of households, it gives the total income of all households, which the median cannot do.

The modal class. The bracket $100,000 to $149,999 has the largest percentage (16.7%), but it is five times as wide as the $25,000 to $34,999 bracket. Per $10,000 of income, households are most concentrated between $25,000 and $35,000 (6.8%), so the bracket with the largest frequency is not where the data are most dense. A modal class is only meaningful when the classes have equal widths.

The median class. The percentages sum to 100.1 (because each is rounded to one decimal place), so the median household is at about the 50.05% mark. The running totals reach 45.3% at $75,000 and 57.3% at $100,000, so the median lies in the bracket $75,000 to $99,999. The published median, $83,730, is in that bracket.

Estimating the mean. The first eight mid-interval values are in the table; the open top class needs a representative value to be assumed. Assuming $250,000, and dividing by the total of the percentages as the mean formula requires:

The estimate, about $106,000, is well short of the published $121,000. Turning the calculation round shows what the top class must contain. The representative value $T$ that reproduces the published mean satisfies

So the households earning $200,000 or more must have a mean income of about $341,000 for the overall mean to be $121,000. The Census Bureau's more detailed table puts the actual mean income of this top group at about $349,000, close to the inferred value; the small gap comes from the mid-interval values used for the lower brackets. The very top of the distribution, invisible in the table, therefore carries a large share of the total, and any estimate of the mean depends directly on the value assumed for it.

Limitations. Every $50,000 added to the assumed value for the open class raises the estimated mean by about $8,000, so the estimate is only as good as that assumption. The mid-interval assumption is also weak in the wide brackets, because within each bracket of a decreasing tail more households lie near the lower boundary than the upper one. Grouped data lose exactly the information that determines the mean in a skewed distribution, which is one more reason the median is preferred for income.

Exam focus

Measures of central tendency and dispersion appear on Paper 1 and Paper 2 at both SL and HL, usually for 1 to 6 marks, and often as the opening parts of a longer statistics question that continues with outliers, correlation or a hypothesis test. Almost every paper contains a "use your GDC to find the mean / standard deviation" part. Common command terms are write down (a value straight from the GDC; no working needed), find and calculate (show the substitution or the GDC input), estimate (grouped data), determine, suggest and justify.

The following question is typical of the opening part of a Paper 2 statistics question. A cinema records the number of tickets bought in each transaction at its box office one evening.

The marks in (a) go to the substituted equation, so it must be written out in full before any solving; the GDC can solve it, but "$k = 26$" alone risks losing the method mark if it is wrong. In (b), (c) and (e), the values come straight from one-variable statistics with the frequency list set, and $n = 100$ in the output confirms that $k$ was entered correctly.

Questions on grouped data reward clear reasons. When asked how a change to the table affects the estimated mean or median, state the direction ("increases", "unchanged") and the reason in terms of the formula or the position of the middle value; a direction with no reason earns at most one mark.

Practice questions

1. The numbers of emails received by an office worker on nine days are 12, 7, 9, 15, 7, 10, 7, 13, 11. Write down the mode, and find the median and the mean. [4]

2. The numbers of mobile phones in a group of households are recorded: 6 households have 1 phone, $p$ have 2, 15 have 3, 9 have 4 and 3 have 5. The mean is 2.75. Find the value of $p$. [3]

3. The masses, $m$ grams, of 50 apples are grouped as follows: $100 \le m < 120$: 4; $120 \le m < 140$: 11; $140 \le m < 160$: 17; $160 \le m < 180$: 12; $180 \le m < 200$: 6. (a) Write down the modal class. (b) Find an estimate of the mean mass. (c) Find an estimate of the standard deviation. [5]

4. For the data in question 1, use your GDC to find the standard deviation and the variance, and explain which of the two standard deviations displayed by the GDC you used. [3]

5. A test is marked out of 60. The mean mark is 38.4 and the standard deviation is 7.2. Each mark $x$ is converted to a final score $y = 1.5x + 10$. Find the mean, the standard deviation and the variance of the final scores. [4]

6. The distances run by a club's members have mean 12.5 miles and variance 9 square miles. Distances are converted to kilometres by $K = \frac{8}{5}M$. Find the mean, the variance and the standard deviation of the distances in kilometres. [4]

7. A student's GDC gives $Q_1 = 6.5$ for the data 4, 6, 7, 9, 12, 15, 18, 21, but a spreadsheet gives 6.75. Show how each value arises, and explain why both can be accepted. [4]

8. Five plants, A to E, are listed in order of height. The median height is 30 cm, the mean is 31 cm and the range is 40 cm. Plant B is 24 cm and plant D is 36 cm tall. Find the heights of plants A and E. [4]

9. Class P has 20 students with a mean score of 64; class Q has 25 students with a mean score of 70. Calculate the mean score of all 45 students, and explain why it is not $\frac{64 + 70}{2}$. [3]

10. The house prices in a small street are $210,000, $225,000, $230,000, $240,000, $245,000, $260,000 and $1,450,000. Suggest, with a reason, which measure of central tendency should be quoted as the typical price, and state the effect on the mean and the median if the most expensive house is removed. [4]

Summary

• The mean is $\bar{x} = \frac{\sum f_i x_i}{n}$ with $n = \sum f_i$: total of the data divided by the number of data items, never by the number of rows. It uses every value and is pulled by outliers.

• The median is the middle value, at position $\frac{n+1}{2}$; the mode is the most frequent value and the only average for qualitative data.

• The median and IQR are robust; the mean, range and standard deviation are not. Use the median for skewed data or data with outliers, and the mean for roughly symmetric data.

• A given mean gives the equation $\sum f_i x_i = n\bar{x}$; with two unknowns, add the total-frequency equation and solve the system.

• For grouped data, the estimated mean uses mid-interval values; the modal class is the interval with the highest frequency (equal widths only), written as an interval.

• Variance is the mean of the squared deviations; standard deviation is its square root, in the data's units. Both come from the GDC: use $\sigma_x$, and square the unrounded value for the variance.

• Quartiles of discrete data come from the GDC; different rules (median of halves, interpolation) can give slightly different values, and all are legitimate.

• For $y = ax + b$ ($a > 0$): measures of position become $a \times \text{old} + b$; standard deviation, range and IQR become

• Compare data sets with one measure of centre and one measure of spread, in context.

Self-check

• To do:

  I can calculate a mean from a frequency table using $\sum f_i x_i$ and $\sum f_i$.

• To do:

  I can find the median of raw data and of a frequency table using the position $\frac{n+1}{2}$ and running totals.

• To do:

  I can write down the mode, and recognise bimodal data.

• To do:

  I can set up and solve an equation for an unknown frequency from a given mean.

• To do:

  I can solve for two unknowns using the mean and the total frequency.

• To do:

  I can explain, with a reason in context, whether the mean or the median is more suitable.

• To do:

  I can find mid-interval values for continuous and discrete classes, and estimate the mean of grouped data.

• To do:

  I can write down a modal class as an interval, and explain why it needs equal class widths.

• To do:

  I can explain why the deviations from the mean sum to zero and why the variance squares them.

• To do:

  I can enter values and frequencies in one-variable statistics and choose $\sigma_x$, not $s_x$.

• To do:

  I can find the variance by squaring the unrounded standard deviation.

• To do:

  I can find quartiles and the IQR with my GDC and explain why another method may give different quartiles.

• To do:

  I can find the new mean, median, standard deviation and variance after a change $y = ax + b$ without recalculating from the data.

• To do:

  I can compare two data sets using both a measure of centre and a measure of spread, in context.

1. (a)

   Write down the value of

   (i) the lower quartile;

   (ii) the upper quartile.

   [2]2 marks

   Markscheme (a)

   (i) $Q_1 = 55$ (bpm) A1

   (ii) $Q_3 = 63$ (bpm) A1

   [2 marks]

2. (b)

   Show that the value 82 bpm is an outlier.

   [3]3 marks

   Markscheme (b)

   $\text{IQR} = 63 - 55 = 8$

   $63 + 1.5 \times 8$ (M1)

   $= 75$

3. (c)

   The scientist later finds that the reading of 82 bpm was taken from a rower who had just finished a warm-up. State, with a reason, whether this value should be removed from the data.

   [1]1 mark

   Markscheme (c)

   it should be removed, because it is not a resting heart rate (the reading does not measure what the study is recording) R1

   [1 mark]

$82 > 75$, so 82 is an outlier R1

Note: The comparison of 82 with 75 must be seen for $\mathbf{R1}$.

[3 marks]