SL 4.3 Calculate and interpret measures of centre, position, and dispersion | International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Statistics and Probability

Summarising data and interpreting what the numbers show

A data set is a collection of observed values. Measures of centre describe a typical value; measures of position describe where values lie in an ordered list; and measures of dispersion describe how spread out the values are. A summary is useful, but it does not show every detail: two groups can have the same centre and different spreads. First identify what each value represents and its units. Then order the data when position matters, choose suitable measures, and explain what the results mean in context. A calculator can check arithmetic and display a box plot, but you should be able to explain what each result represents.

What you will learn

Calculate and interpret the mean, median, and mode.
Find quartiles, the interquartile range, and the range.
Calculate and interpret the population standard deviation of a complete data set.
Use numerical summaries and box plots to compare data, and check calculations with graphing technology.

1. Prior knowledge and measures of centre

The number of observations in a data set is denoted by

n

. To calculate a mean, add all observations and divide by

n

. In a frequency table, multiply each value by its frequency, add those products, and divide by the total frequency.

The mean, written as

\bar{x}

, uses every observation. The median is the middle value after the data are placed in increasing order. If there is an even number of observations, the median is the mean of the two middle values. The mode is the value, or values, that occurs most often; a data set may have one mode, several modes, or no mode.

A very high or low observation can pull the mean toward it, while the median depends on ordered position. Both have the same units as the data. The mode is useful when the most common recorded value matters, including for data recorded in categories.

\bar{x}=\frac{\sum_{i=1}^{n}x_i}{n}

Order the data before finding the median or quartiles.
The mean uses every observation; the median is less affected by an extreme value.
Include units and context when interpreting a statistic.

2. Position and graphical summaries

Quartiles describe positions in ordered data. The lower quartile,

Q_1

, and upper quartile,

Q_3

, mark positions around which the lower and upper quarters of the data lie. The median is also called

Q_2

. A common hand method, when there is an odd number of observations, is to exclude the overall median and find the medians of the lower and upper halves. Calculator conventions can differ, so follow the method stated in a question or state the method you use.

A box plot displays the minimum,

Q_1

, median,

Q_3

, and maximum on a number line. The box extends from

Q_1

to

Q_3

and represents the middle half of the data; the line inside marks the median. In a standard five-number-summary box plot, whiskers extend to the minimum and maximum. A box plot helps compare centres and spreads, but does not show every individual value.

To check results with a graphing calculator, enter the observations in a list, confirm that the list contains the intended values, and display a box plot or summary statistics. Check that the scale and units make sense. If the question specifies a quartile convention, compare the calculator output with that convention.

Q_1\leq Q_2\leq Q_3

Quartiles describe locations in ordered data, not fixed fractions of the numerical scale.
The box in a box plot represents the middle half of the data.
Check the quartile convention when using calculator output.

3. Measures of dispersion

The range is the maximum value minus the minimum value. It describes the full span of the observations, but depends only on the two extremes. The interquartile range, abbreviated IQR, is the upper quartile minus the lower quartile. It describes the width of the middle half and is less affected by extreme values.

Standard deviation describes spread around the mean. To find it for a complete data set, calculate each value's difference from the mean, square the differences, average the squares, and take the square root. Squaring prevents differences above and below the mean from cancelling. The population standard deviation is written as

\sigma

and has the same units as the data. A larger standard deviation means greater spread around the mean; it does not mean a larger mean.

When a task asks for the standard deviation of a complete data set, choose the population standard deviation on the calculator. Calculator labels vary, so check the selected statistic, entered list, and units. Treat the display as a calculation to interpret, not as an explanation.

\mathrm{IQR}=Q_3-Q_1

Range measures the full span; IQR measures the spread of the middle half.
Standard deviation is non-negative and uses the original data units.
Use a centre measure and a spread measure together when comparing groups.

4. Choosing, comparing, and communicating summaries

Choose summaries that suit the data and the question. If a data set has an extreme value or is strongly uneven, the median and IQR can describe its centre and middle spread without being pulled as strongly by that extreme. For reasonably balanced data without influential extremes, the mean and standard deviation can be informative together because both use all observations.

In a contextual comparison, name the statistic and explain what its size means. A higher median journey time means the middle journey time is longer. A larger IQR means the middle half of journey times covers a wider interval. A larger range means the full span is wider; a larger standard deviation means greater spread around the mean.

For a complete response, show the requested calculation or method, give units, and compare the results in words. A graphing calculator can check arithmetic and display a box plot, but your reasoning should still explain what the results say about the data.

Do not infer spread from a centre measure, or centre from a spread measure.
A contextual comparison explains what the numerical difference means.
A box plot supports comparison but does not show every data value.

What the measures describe

Measure	What it describes	Useful caution
Mean	Centre using all values	Can be pulled by extreme values
Median	Middle of ordered values	Depends on position, not every value's size
Range	Full span from minimum to maximum	Depends only on the extremes
IQR	Width of the middle half	Does not describe the extremes
Standard deviation	Spread around the mean	Interpret alongside a centre measure

Worked example

Finding centre and position

The numbers of books read by seven students in a month are 2, 4, 5, 5, 7, 8, 13. Find the mean, median, mode, and quartiles using the median-of-halves convention.

Check the order
The values are already in increasing order. With seven observations, the fourth value is the median.
$2,\ 4,\ 5,\ 5,\ 7,\ 8,\ 13$
Calculate the mean
Add the seven values and divide by seven. The mean need not be a whole number even though the individual book counts are whole numbers.
$\bar{x}=\frac{44}{7}\approx 6.29$
Identify the median, mode, and quartiles
The fourth value is the median, and $5$ occurs most often. Excluding the overall median leaves lower values CAD 2, 4, 5 and upper values CAD 7, 8, 13. The medians of these two halves give the quartiles.
$Q_1=4,\quad Q_2=5,\quad Q_3=8$

Answer: The mean is approximately 6.29 books, the median is 5 books, and the mode is 5 books. The quartiles are

Q_1=4

,

Q_2=5

, and

Q_3=8

books.

Check: The mean is greater than the median because the high value of 13 pulls the mean upward.

Worked example

Comparing spread in context

Two teams record practice times in minutes. Team A has minimum 12, lower quartile 15, median 18, upper quartile 21, and maximum 26. Team B has minimum 10, lower quartile 16, median 19, upper quartile 22, and maximum 24. Compare their centres and spreads.

Compare the medians
Team B's median is one minute greater, so its middle recorded practice time is slightly longer.
$19-18=1$
Calculate the interquartile ranges
Subtract the lower quartile from the upper quartile for each team. This compares the widths of their middle halves.
$\mathrm{IQR}_A=21-15=6,\quad \mathrm{IQR}_B=22-16=6$
Calculate the ranges
Subtract each minimum from its maximum to compare the full span of recorded times.
$\mathrm{Range}_A=26-12=14,\quad \mathrm{Range}_B=24-10=14$

Answer: Team B's median is one minute higher. Both teams have an IQR of 6 minutes and a range of 14 minutes. These measures show equal middle-half widths and equal full spans, although the quartiles and extreme values are not identical.

Check: A box plot would show equal box widths and equal full spans, with Team B's median one minute farther to the right.

Worked example

Calculating population standard deviation

A complete set of three plant-height measurements, in centimetres, is 4, 5, 6. Find the population standard deviation to two decimal places.

Find the mean
Add the measurements and divide by the three observations.
$\bar{x}=\frac{4+5+6}{3}=5$
Find the squared deviations
Subtract the mean from each measurement and square each difference. The squared differences are non-negative whether a measurement is above or below the mean.
$\sum_{i=1}^{3}(x_i-5)^2=(4-5)^2+(5-5)^2+(6-5)^2=2$
Average and take the square root
These measurements are the complete data set, so divide the sum of squared deviations by three and take the square root. Round only the final result to two decimal places.
$\sigma=\sqrt{\frac{2}{3}}\approx 0.82$

Answer: The population standard deviation is approximately 0.82 cm.

Check: Each measurement is at most 1 cm from the mean, so a standard deviation below 1 cm is reasonable.

Common mistakes and how to avoid them

Finding the median or quartiles before ordering the data.

Correction: Put the observations in increasing order first, because these measures depend on position in the ordered list.

Calling the difference between the quartiles the range.

Correction: The difference

Q_3-Q_1

is the IQR. The range is maximum minus minimum.

Interpreting a larger standard deviation as a larger mean.

Correction: Standard deviation describes spread around the mean, not the location of the centre.

Reporting calculator output without units or context.

Correction: Name the measure, include its units, and explain what it says about the data.

Lesson summary

Mean, median, and mode describe centre in different ways.
Quartiles describe positions in ordered data; a box plot displays the five-number summary.
Range, IQR, and standard deviation describe different aspects of dispersion.
Select suitable summaries, check calculations, and interpret results in context.

Check your understanding

Question 1

For the ordered data 3, 5, 5, 9, what is the median?

5
6
5.5
22

Show answer and explanation

5

There are four values, so average the two middle values:

(5+5)/2=5

.

Question 2

A data set has

Q_1=12

and

Q_3=20

. What is its IQR?

8
32
20
12

Show answer and explanation

8

Subtract the lower quartile from the upper quartile:

20-12=8

.

Question 3

Two groups have the same mean, but Group X has a larger standard deviation. What does this indicate?

Group X has a higher centre.
Group X's values are more spread around the mean.
Group X has a smaller range for certain.
Every value in Group X is larger.

Show answer and explanation

Group X's values are more spread around the mean.

Standard deviation measures spread around the mean, so the larger value indicates greater spread. It does not determine the range or the ordering of every observation.

Key terms

Mean: The sum of the data values divided by the number of values.
Median: The middle value in ordered data, or the mean of the two middle values when there is an even number.
Quartile: A value marking a position in ordered data that divides it into approximately four parts.
Interquartile range: The upper quartile minus the lower quartile; it describes the width of the middle half of the data.
Standard deviation: A measure of spread around the mean, expressed in the original data units.

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SL 4.3 · Calculate and interpret measures of centre, position, and dispersion

International Baccalaureate (IB) IB AA SL: Mathematics: Analysis and Approaches SL

Summarising data and interpreting what the numbers show

What you will learn

1. Prior knowledge and measures of centre

2. Position and graphical summaries

3. Measures of dispersion

4. Choosing, comparing, and communicating summaries

What the measures describe

Finding centre and position

Comparing spread in context

Calculating population standard deviation

Common mistakes and how to avoid them

Lesson summary

Check your understanding

Question 1

Question 2

Question 3

Key terms

Continue through IB AA SL

About this lesson and its review