Collect data and calculate summaries

A shirt number is used only to identify a player. Does averaging the shirt numbers describe a numerical measurement of the players?

Ask how the values arise

A variable is the attribute recorded for each observation. Numerical variables describe quantities for which arithmetic has a meaning

Discrete numerical values arise as separate countable possibilities, such as the number of counters in a container

Continuous numerical values come from measurement on a scale, such as length or elapsed time

Measurement records have limited precision. A length recorded as 8 cm is still a measurement of an underlying continuous quantity

A category may be written as a word or an identifying number. That does not make it a numerical measurement

For each seedling, a student counts its individual leaves. What type of numerical variable is recorded?

A student measures the length of each fallen leaf with a ruler, recording centimetres to the nearest tenth. What type of numerical variable is its length?

Keep a clear acquisition record

Choose the variable, the objects or events to observe, and the counting or measuring procedure before collecting data

Record one value per observation, with a label, unit and consistent measurement precision

Keep repeated values: two objects with the same measurement are still two observations

A missing measurement is not automatically zero. A genuine zero count is a recorded value and remains in the data

The supplied calculation data are illustrative. For a real investigation, retain the original records and note any measurement problem before considering a correction

Five measured lengths are recorded as 6, 7, 7, 8 and 9 cm to the nearest centimetre. Which conclusion follows?

Mean is the equal-share value

Add every recorded observation, including repeats and genuine zeros

Divide that total by the number of observations. The mean is the amount each would have if the total were shared equally

The symbol x-bar names the mean. A mean need not equal any individual observation

A mean count can be fractional even when every individual count is a whole number. It describes an average, not part of one physical object

Add all five observations before dividing

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 4, 7, 3, 6, 5

Calculate the mean of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Use the number of observations as the divisor

Count repeated observations separately

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 2, 3, 3, 5, 7

Calculate the mean of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Keep all six entries in the total

Keep an exact mean when requested

Division may give a recurring decimal. An exact simplified fraction avoids rounding it

Ten thirds is exact; 3.33 is only an approximation

When the question asks for an exact value, give a whole number, an exact terminating decimal or a simplified fraction. Round only when a recording or reporting instruction requests it

Retain a recurring mean as a fraction

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 2, 3, 5

Calculate the mean of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Add all 3 recorded values

Divide by the number of observations, not the number of different values

Keep the exact fraction rather than rounding its recurring decimal

Distinguish the exact mean from a rounded decimal

Average recorded measurements in one unit

Illustrative data record the length of each fallen leaf, measured in centimetres to the nearest tenth

These are recorded measurements of an underlying continuous quantity. Each entry is one observation

Data: 4.2, 5.1, 4.8, 5.4, 5

Calculate the mean of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Use decimal observations without losing precision

Order the observations before taking the middle

The median is determined by position in the ordered data, not by the original recording order

For an odd number of observations, the median is the single middle value

With five observations, two lie on each side of position three

Repeated values each occupy their own position

Locate the third value after sorting

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 9, 4, 6, 5, 8

Calculate the median of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Order the observations and locate the middle position or positions

Order the data: 4, 5, 6, 8, 9

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Ignore the middle of the unsorted record

Average the two middle values for an even count

For an even number of observations, there is no single middle position

After ordering, use positions n divided by two and one position after it

Add those two data values and divide by two. Do not average the position numbers

The median can fall between two observations and need not appear in the list

Calculate between the two middle observations

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 4, 9, 3, 7, 5, 8

Calculate the median of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Order the observations and locate the middle position or positions

Order the data: 3, 4, 5, 7, 8, 9

There are 6 values, so use positions 3 and 4

The median depends on ordered position, not the original recording order

Allow a half-integer median of count data

Keep equal middle values in separate positions

Average two decimal middle values exactly

Illustrative data record the length of each fallen leaf, measured in centimetres to the nearest tenth

These are recorded measurements of an underlying continuous quantity. Each entry is one observation

Data: 4.1, 3.7, 4.8, 4.2

Calculate the median of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Order the observations and locate the middle position or positions

Order the data: 3.7, 4.1, 4.2, 4.8

There are 4 values, so use positions 2 and 3

The median depends on ordered position, not the original recording order

The exact median may have another decimal place

Range measures the distance between extremes

Find the largest and smallest observed values, then subtract smallest from largest

Keep the same measurement unit as the data. The range is a measure of spread, not a measure of centre

The first and last recorded entries are not necessarily the smallest and largest

If every observation is equal, the range is zero even when there are many observations

Use actual extremes rather than recording endpoints

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 9, 4, 6, 5, 8

Calculate the range of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use the smallest and largest observed values

The largest value is 9 and the smallest value is 4

The range describes the distance between extremes, not a central value

Subtract the minimum from the maximum

Keep the measurement unit in a decimal range

Illustrative data record the length of each fallen leaf, measured in centimetres to the nearest tenth

These are recorded measurements of an underlying continuous quantity. Each entry is one observation

Data: 4.2, 5.1, 4.8, 5.4, 5

Calculate the range of the supplied data

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use the smallest and largest observed values

The largest value is 5.4 and the smallest value is 4.2

The range describes the distance between extremes, not a central value

Calculate the exact spread of recorded lengths

Recognise zero spread in a constant record

Mode names the most frequent value

Count how many times each different data value appears. This count is its frequency

A mode is a value with the greatest frequency, not the frequency number itself

Use every value tied for greatest repeating frequency. Two such values give two modes; more than two are also possible

In this topic, if every value occurs once, report no mode. This convention is stated in each mode question

Do not take the mean of two modes: report both observed modal values

Separate the repeated value from its frequency

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 4, 6, 4, 8, 5

Use every value tied for greatest frequency as a mode when that frequency exceeds one. If every value occurs once, report no mode

Identify every mode, or select No mode when no value repeats

Count every occurrence, including repeats

4 occurs 2 times

5 occurs 1 time

6 occurs 1 time

8 occurs 1 time

The greatest frequency is 2, so the mode is 4

A mode is a data value, not its frequency

Count all repetitions

Retain both values tied at the greatest frequency

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 2, 4, 2, 6, 4, 7

Use every value tied for greatest frequency as a mode when that frequency exceeds one. If every value occurs once, report no mode

Identify every mode, or select No mode when no value repeats

Count every occurrence, including repeats

2 occurs 2 times

4 occurs 2 times

6 occurs 1 time

7 occurs 1 time

The greatest frequency is 2. Every value tied at that frequency is a mode: 2, 4

A mode is a data value, not its frequency

Select the complete set of modes

Use the stated no-mode convention

A tie can involve more than two values

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 1, 1, 3, 3, 5, 5

Use every value tied for greatest frequency as a mode when that frequency exceeds one. If every value occurs once, report no mode

Identify every mode, or select No mode when no value repeats

Count every occurrence, including repeats

1 occurs 2 times

3 occurs 2 times

5 occurs 2 times

The greatest frequency is 2. Every value tied at that frequency is a mode: 1, 3, 5

A mode is a data value, not its frequency

Keep the four operations distinct

Mean uses the total and the number of observations. Median uses the middle ordered position or positions

Mode uses frequencies. Range uses the two extreme values

They can coincide numerically, but that does not make their meanings interchangeable

Calculate from the same complete data set and label each result so the reader knows what it describes

Build a complete summary record

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 3, 4, 4, 5, 9

Calculate the four requested summaries of this data set

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 3, 4, 4, 5, 9

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Count every occurrence, including repeats

3 occurs 1 time

4 occurs 2 times

5 occurs 1 time

9 occurs 1 time

The greatest frequency is 2, so the mode is 4

A mode is a data value, not its frequency

Use the smallest and largest observed values

The largest value is 9 and the smallest value is 3

The range describes the distance between extremes, not a central value

Different summaries can happen to be equal

Summarise six measured lengths

Illustrative data record the length of each fallen leaf, measured in centimetres to the nearest tenth

These are recorded measurements of an underlying continuous quantity. Each entry is one observation

Data: 4.2, 4.5, 4.5, 5.1, 5.4, 6.3

Calculate the four requested summaries of this data set

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Use every observation in the total, including repeated values

Add the 6 displayed values

There are 6 values, so divide the sum by 6

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 4.2, 4.5, 4.5, 5.1, 5.4, 6.3

There are 6 values, so use positions 3 and 4

The median depends on ordered position, not the original recording order

Count every occurrence, including repeats

4.2 occurs 1 time

4.5 occurs 2 times

5.1 occurs 1 time

5.4 occurs 1 time

6.3 occurs 1 time

The greatest frequency is 2, so the mode is 4.5

A mode is a data value, not its frequency

Use the smallest and largest observed values

The largest value is 6.3 and the smallest value is 4.2

The range describes the distance between extremes, not a central value

Keep each exact decimal summary separate

Investigate an unusual observation without automatically deleting it

A value far from the main group may strongly affect some summaries. It may be a genuine observation, a recording error or a measurement problem

Check the original record and context before deciding what to do with it

For a requested comparison, calculate summaries with all values and then with only the specified value omitted

The mean uses every observation and can be pulled towards an extreme. The range directly uses an extreme; the median depends on the middle positions

A median is not guaranteed to remain unchanged when an observation is removed. Recalculate it from the remaining ordered list

The first four values are 8, 9, 9 and 10. Move the fifth observation through its five settings from 10 to 30. Compare the mean, median and range each time

What happens in this particular five-value family as the fifth observation increases?

A removed value can also change the median

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 8, 9, 10, 11, 24

For comparison only, omit the observation 24 at recorded position 5. Keep every other observation

This calculation does not establish that the separated value is an error or should be deleted from the original record

Calculate mean, median and range before and after the specified omission

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

First calculate with all 5 observations

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 8, 9, 10, 11, 24

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 24 and the smallest value is 8

The range describes the distance between extremes, not a central value

Omit only the specified observation. The remaining data are 8, 9, 10, 11

Use every observation in the total, including repeated values

Add the 4 displayed values

There are 4 values, so divide the sum by 4

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 8, 9, 10, 11

There are 4 values, so use positions 2 and 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 11 and the smallest value is 8

The range describes the distance between extremes, not a central value

Mean changes from 12.4 to 9.5; the size of the change is 2.9

The mean uses every observed value, so a separated observation can pull it towards that extreme

Median changes from 10 to 9.5; the size of the change is 0.5

Check the middle positions again: the median is not guaranteed to stay unchanged in every removal

Range changes from 16 to 3; the size of the change is 13

An extreme observation directly affects the range

Omit only the stated observation for the comparison

An extreme can also lie below the main group

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data: 2, 14, 15, 15, 16

For comparison only, omit the observation 2 at recorded position 1. Keep every other observation

This calculation does not establish that the separated value is an error or should be deleted from the original record

Calculate mean, median and range before and after the specified omission

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

First calculate with all 5 observations

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 2, 14, 15, 15, 16

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 16 and the smallest value is 2

The range describes the distance between extremes, not a central value

Omit only the specified observation. The remaining data are 14, 15, 15, 16

Use every observation in the total, including repeated values

Add the 4 displayed values

There are 4 values, so divide the sum by 4

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 14, 15, 15, 16

There are 4 values, so use positions 2 and 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 16 and the smallest value is 14

The range describes the distance between extremes, not a central value

Mean changes from 12.4 to 15; the size of the change is 2.6

The mean uses every observed value, so a separated observation can pull it towards that extreme

Median changes from 15 to 15; the size of the change is 0

Check the middle positions again: the median is not guaranteed to stay unchanged in every removal

Range changes from 14 to 2; the size of the change is 12

An extreme observation directly affects the range

Recalculate after omitting one low value

Choose a centre for the question being asked

Use the mean when the equal-share or total-per-observation value is the purpose. Every observation contributes

Use the median when a middle observation is useful and the centre should be less affected by a distant value

Use the mode when the most common recorded value is the relevant information

In a roughly symmetric distribution without unusual extremes, mean and median can give similar centre descriptions. With a long tail, the mean may be pulled towards it

Give the purpose, the numerical summaries and the relevant data pattern when explaining a choice. No single centre is always best

Illustrative task times are 8, 9, 9, 10 and 24 seconds. Their mean is 12 seconds and median is 9 seconds. The aim is to describe a middle task time without the separated 24-second observation pulling the centre upward. Which choice fits?

Illustrative counts in five containers are 2, 4, 4, 7 and 8 counters. The question asks how many counters each container would hold if all twenty-five were shared equally. Which summary answers that question?

Recorded leaf counts are 3, 5, 3, 7, 3 and 8. The question asks which count occurred most often. Which summary answers it?

Matching centres do not identify the whole data set

Two different data sets can have the same mean and median

Their spread can still differ, so inspect a measure such as range and the original observations

A summary deliberately discards detail. Do not reconstruct the entire distribution from one average

When comparing sets, keep the variable and units consistent and calculate each set separately

Compare two sets with matching centres

Illustrative data record the number of leaves on each seedling

These are counts of individual leaves. Each entry is one observation

Data set A: 4, 5, 6, 7, 8

Data set B, recorded for the same variable and unit: 2, 4, 6, 8, 10

Calculate mean, median and range separately for sets A and B

Give exact values as whole numbers, terminating decimals or simplified fractions. Do not round

Calculate each summary for set A

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 4, 5, 6, 7, 8

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 8 and the smallest value is 4

The range describes the distance between extremes, not a central value

Now calculate the same summaries for set B

Use every observation in the total, including repeated values

Add the 5 displayed values

There are 5 values, so divide the sum by 5

The mean is the equal-share value for this total and number of observations

Order the observations and locate the middle position or positions

Order the data: 2, 4, 6, 8, 10

There are 5 values, so use position 3

The median depends on ordered position, not the original recording order

Use the smallest and largest observed values

The largest value is 10 and the smallest value is 2

The range describes the distance between extremes, not a central value

The means and medians match, but the ranges differ. Equal centres do not determine the complete data set or its spread

Calculate spread as well as the shared centre

Acquire and summarise your own numerical data

Written collection task

Make eight small piles of counters and record the count in each pile as one discrete data set

Separately, choose eight fallen leaves and measure each length from base to tip in centimetres to the nearest tenth. Record the measuring rule and each value as a continuous data set

Keep observations labelled and retain repeats. Mark any missing measurement as missing rather than entering zero

Calculate mean, median, mode and range for each complete record. For mode, report every value tied for the greatest frequency if that frequency exceeds one; otherwise report no mode. Keep the two different variables separate

Check any unusual value against the original count or measurement. Explain which centre is useful for a stated purpose and what information that centre leaves out

Use definitions and context together

Determine whether a numerical variable is counted or measured, and keep a consistent acquisition record

Mean uses all observations and their count. Median uses the middle ordered position or positions

Mode uses the greatest repeating frequency under the stated convention. Range is maximum minus minimum

Retain repeats and exact values when requested; do not replace missing data with zero

Investigate unusual observations rather than automatically deleting them

Choose a centre for the purpose and explain the relevant pattern. Matching centres can hide different spreads