Construct and compare numerical data displays

The recorded leaf counts are 2, 3, 3 and 5. How many dots should a dot plot contain?

Build a numerical scale before stacking dots

A dot plot places observations on a numerical horizontal scale. One dot represents one observation

Label the variable and its unit, and use equal distances for equal numerical differences

Stack repeated observations above the same value. The stack height is the frequency, not the observation value

Keep unused values in their proper positions on the scale. A gap has frequency zero

For the construction records here, use the smallest and largest observed values as the axis endpoints and give every listed stack height

Place every observation once

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Data: 2, 3, 3, 4, 5, 5, 6

Construct the complete stack-height record using every listed axis position

Enter exact values, with zero for an empty stack. Do not round

Order the data without deleting repeats: 2, 3, 3, 4, 5, 5, 6

Use a regular scale from 2 to 6 in steps of 1

2 occurs 1 time, so draw 1 dot above it

3 occurs 2 times, so draw 2 dots above it

4 occurs 1 time, so draw 1 dot above it

5 occurs 2 times, so draw 2 dots above it

6 occurs 1 time, so draw 1 dot above it

Check the total: the stacks contain 7 dots for 7 observations

Complete every stack height

A missing stack is not a missing scale position

If values two and four occur but three does not, keep the equally spaced position for three

No dot is drawn above that position. Its frequency is zero

Do not squeeze the occupied positions together as though the values were unordered categories

Check the sum of all stack heights against the number of original observations

Retain a zero-frequency position

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Data: 2, 2, 4, 5, 5, 6

Construct the complete stack-height record using every listed axis position

Enter exact values, with zero for an empty stack. Do not round

Order the data without deleting repeats: 2, 2, 4, 5, 5, 6

Use a regular scale from 2 to 6 in steps of 1

2 occurs 2 times, so draw 2 dots above it

3 occurs 0 times, so draw 0 dots above it

4 occurs 1 time, so draw 1 dot above it

5 occurs 2 times, so draw 2 dots above it

6 occurs 1 time, so draw 1 dot above it

Check the total: the stacks contain 6 dots for 6 observations

Keep both gaps on the scale

Measured data still need an evenly spaced scale

A measured variable can be continuous even when its record is rounded to tenths

Use the stated unit and precision when marking the numerical axis

A step from 4.1 to 4.2 has the same size as a step from 4.2 to 4.3

Repeated recorded measurements still contribute separate dots

Construct a tenths-scale display

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

One dot represents one observation, including repeated values

Data: 4.1, 4.1, 4.3, 4.4, 4.4, 4.6

Construct the complete stack-height record using every listed axis position

Enter exact values, with zero for an empty stack. Do not round

Order the data without deleting repeats: 4.1, 4.1, 4.3, 4.4, 4.4, 4.6

Use a regular scale from 4.1 to 4.6 in steps of 0.1

4.1 occurs 2 times, so draw 2 dots above it

4.2 occurs 0 times, so draw 0 dots above it

4.3 occurs 1 time, so draw 1 dot above it

4.4 occurs 2 times, so draw 2 dots above it

4.5 occurs 0 times, so draw 0 dots above it

4.6 occurs 1 time, so draw 1 dot above it

Check the total: the stacks contain 6 dots for 6 observations

Keep the decimal scale and zero stacks

Recover the data before calculating summaries

Read one horizontal value for every dot, repeating a value once for each dot in its stack

The total number of dots is the number of observations. The tallest stack locates a modal value, not the mean

Mean uses the sum of all observed values divided by the number of dots

For the median, read the observations in numerical order and locate the middle position or positions

Range is the largest observed value minus the smallest. It is not the number of tick marks

Read a gap and all four summaries

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Read the plotted observations and complete the requested summaries

Enter exact values, with zero for an empty stack. Do not round

Read one value for each dot: 2, 3, 3, 4, 7

There are 5 dots in total; 5 has frequency 0

The total of the observations is 19. Divide by 5 to obtain mean 3.8

The middle position is 3; median 3

Range is 7 minus 2, giving 5

The highest frequency is 2, at 3. That data value is the mode

All results refer to leaves per seedling

Count dots rather than occupied positions

Decode measurements from the plot

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

One dot represents one observation, including repeated values

Read the plotted observations and complete the requested summaries

Enter exact values, with zero for an empty stack. Do not round

Read one value for each dot: 3.2, 3.3, 3.3, 3.4, 3.6, 3.7

There are 6 dots in total; 3.5 has frequency 0

The total of the observations is 20.5. Divide by 6 to obtain mean 41/12

The middle positions are 3 and 4, with values 3.3 and 3.4; median 3.35

Range is 3.7 minus 3.2, giving 0.5

The highest frequency is 2, at 3.3. That data value is the mode

All results refer to leaf length (cm)

Separate a decimal mode from its frequency

Use a key to split each value into stem and leaf

An ordered stem-and-leaf plot preserves the individual observed values

The key states the place value. With the key 2 | 4 means 24 seeds, the stem is the leading digit and the leaf is the units digit

Write stems in increasing order, including intermediate stems with no observations

Within each row, write the leaves in increasing order. Keep repeated leaves and a zero leaf when an observed value ends in zero

An empty row has no leaf. Writing a zero leaf would create an observation that was not collected

A stem plot has the key 3 | 7 means 3.7 cm. What does the entry 4 | 2 represent?

Construct every ordered row

Illustrative data record the number of seeds in each tray

Seed counts are discrete whole-number observations

Key: 2 | 1 means 21 seeds

Data: 21, 22, 22, 24, 30, 31

Use every consecutive stem from 2 to 3, including empty rows

Construct the complete ordered stem-and-leaf record using every listed row

Enter each leaf as one digit and each row count as a whole number, using count zero for an empty row

Order all observations without deleting repeats: 21, 22, 22, 24, 30, 31

Use the key 2 | 1 means 21 seeds

21 belongs to stem 2 with leaf 1

22 belongs to stem 2 with leaf 2

22 belongs to stem 2 with leaf 2

24 belongs to stem 2 with leaf 4

30 belongs to stem 3 with leaf 0

31 belongs to stem 3 with leaf 1

Row 2 contains 4 leaves, ordered 1, 2, 2, 4

Row 3 contains 2 leaves, ordered 0, 1

The row counts total 6, matching the number of observations

A leaf of zero is an observation, whereas an empty row contains no observations

Retain repeats and a zero leaf

Leave an empty stem row empty

If the smallest value is in the twenties and the largest is in the forties, include stems two, three and four

If no values lie in the thirties, stem three has count zero and no leaves

A row written as 3 | 0 instead records the value thirty under a whole-number key

The total number of leaves across all rows must equal the number of observations

Keep an empty middle stem

Illustrative data record the number of seeds in each tray

Seed counts are discrete whole-number observations

Key: 2 | 2 means 22 seeds

Data: 22, 22, 24, 41, 43

Use every consecutive stem from 2 to 4, including empty rows

Construct the complete ordered stem-and-leaf record using every listed row

Enter each leaf as one digit and each row count as a whole number, using count zero for an empty row

Order all observations without deleting repeats: 22, 22, 24, 41, 43

Use the key 2 | 2 means 22 seeds

22 belongs to stem 2 with leaf 2

22 belongs to stem 2 with leaf 2

24 belongs to stem 2 with leaf 4

41 belongs to stem 4 with leaf 1

43 belongs to stem 4 with leaf 3

Row 2 contains 3 leaves, ordered 2, 2, 4

Row 3 contains 0 leaves, so its count is zero and no leaf is written

Row 4 contains 2 leaves, ordered 1, 3

The row counts total 5, matching the number of observations

A leaf of zero is an observation, whereas an empty row contains no observations

Distinguish an empty row from a zero leaf

Construct an ordered plot of measured lengths

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

Lengths are recorded measurements of an underlying continuous quantity

Key: 2 | 1 means 2.1 cm

Data: 2.1, 2.2, 2.2, 2.4, 3.0, 3.1

Use every consecutive stem from 2 to 3, including empty rows

Construct the complete ordered stem-and-leaf record using every listed row

Enter each leaf as one digit and each row count as a whole number, using count zero for an empty row

Order all observations without deleting repeats: 2.1, 2.2, 2.2, 2.4, 3.0, 3.1

Use the key 2 | 1 means 2.1 cm

2.1 belongs to stem 2 with leaf 1

2.2 belongs to stem 2 with leaf 2

2.2 belongs to stem 2 with leaf 2

2.4 belongs to stem 2 with leaf 4

3.0 belongs to stem 3 with leaf 0

3.1 belongs to stem 3 with leaf 1

Row 2 contains 4 leaves, ordered 1, 2, 2, 4

Row 3 contains 2 leaves, ordered 0, 1

The row counts total 6, matching the number of observations

A leaf of zero is an observation, whereas an empty row contains no observations

Use a decimal key across an empty stem

Count leaves, not rows

Decode every leaf with its stem and the supplied key

Empty rows contribute no observations. Repeated leaves contribute repeated observations

Read rows in increasing stem order and leaves in increasing order to recover an ordered list

Count all leaves before finding the middle position or positions

Use the recovered values for mean and range; leaf digits alone are not the original observations

Recover a full ordered list

Illustrative data record the number of seeds in each tray

Seed counts are discrete whole-number observations

Key: 2 | 1 means 21 seeds

Read the plot and calculate the count, mean, median and range

Give exact whole numbers, terminating decimals or simplified fractions without rounding

Decode each leaf using 2 | 1 means 21 seeds

Row 2 represents 21, 22, 22, 24

Row 3 represents 30, 31

In order, the observations are 21, 22, 22, 24, 30, 31

There are 6 leaves, including each repeated leaf, so there are 6 observations

The total is 150 seeds; dividing by 6 gives mean 25 seeds

The middle positions are 3 and 4, with values 22 and 24; add these two values and divide by 2

The median is 23 seeds

Range is 31 minus 21, giving 10 seeds

Read values across an empty row

Summarise a decimal stem plot

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

Lengths are recorded measurements of an underlying continuous quantity

Key: 4 | 1 means 4.1 cm

Read the plot and calculate the count, mean, median and range

Give exact whole numbers, terminating decimals or simplified fractions without rounding

Decode each leaf using 4 | 1 means 4.1 cm

Row 4 represents 4.1, 4.2, 4.2, 4.4

Row 5 represents 5.0, 5.1

In order, the observations are 4.1, 4.2, 4.2, 4.4, 5.0, 5.1

There are 6 leaves, including each repeated leaf, so there are 6 observations

The total is 27.0 cm; dividing by 6 gives mean 4.5 cm

The middle positions are 3 and 4, with values 4.2 and 4.4; add these two values and divide by 2

The median is 4.3 cm

Range is 5.1 minus 4.1, giving 1.0 cm

Retain a precise median from decimal leaves

Split a crowded stem using an explicit rule

A split stem uses two rows for the same stem to show the distribution more clearly

Here L contains leaf digits zero to four and H contains leaf digits five to nine

Read 2 L before 2 H, then 3 L before 3 H. The key still determines the original value

For example, 24 belongs in 2 L and 25 in 2 H under a whole-number key

An empty half-row contains no leaves. Splitting changes the display arrangement, not the observations or their summaries

Place boundary leaves four and five

Illustrative data record the number of seeds in each tray

Seed counts are discrete whole-number observations

Key: 2 | 4 means 24 seeds

Each stem has two rows: L for leaves 0 to 4 and H for leaves 5 to 9

Data: 24, 25, 30, 37

Use every consecutive stem from 2 to 3, including empty rows

Construct the complete ordered stem-and-leaf record using every listed row

Enter each leaf as one digit and each row count as a whole number, using count zero for an empty row

Order all observations without deleting repeats: 24, 25, 30, 37

Use the key 2 | 4 means 24 seeds

24 belongs to stem 2 in row L with leaf 4

25 belongs to stem 2 in row H with leaf 5

30 belongs to stem 3 in row L with leaf 0

37 belongs to stem 3 in row H with leaf 7

Row 2L contains 1 leaf, ordered 4

Row 2H contains 1 leaf, ordered 5

Row 3L contains 1 leaf, ordered 0

Row 3H contains 1 leaf, ordered 7

The row counts total 4, matching the number of observations

A leaf of zero is an observation, whereas an empty row contains no observations

Complete every split-row count and leaf

Apply split rows to measured lengths

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

Lengths are recorded measurements of an underlying continuous quantity

Key: 2 | 0 means 2.0 cm

Each stem has two rows: L for leaves 0 to 4 and H for leaves 5 to 9

Read the plot and calculate the count, mean, median and range

Give exact whole numbers, terminating decimals or simplified fractions without rounding

Decode each leaf using 2 | 0 means 2.0 cm

Row 2L represents 2.0, 2.4

Row 2H represents 2.5, 2.9

Row 3L represents 3.0, 3.4

Row 3H represents 3.5, 3.8

In order, the observations are 2.0, 2.4, 2.5, 2.9, 3.0, 3.4, 3.5, 3.8

There are 8 leaves, including each repeated leaf, so there are 8 observations

The total is 23.5 cm; dividing by 8 gives mean 2.9375 cm

The middle positions are 4 and 5, with values 2.9 and 3.0; add these two values and divide by 2

The median is 2.95 cm

Range is 3.8 minus 2.0, giving 1.8 cm

Read a split plot without changing its data

Describe tails, peaks, clusters and gaps

A symmetric distribution has similar frequencies at matching distances either side of its centre

Positive or right skew describes a longer tail towards higher values. Negative or left skew describes a longer tail towards lower values

Name the direction of the tail, not the end containing most observations

A cluster is a group of nearby observations; a gap is an interval with no observations

A bimodal display has two prominent peaks. A display can be both bimodal and symmetric

Use these features with numerical summaries. A mean or median alone does not specify the entire shape

Recognise balance around a central peak

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Which complete description matches the distribution’s tails and peaks?

Read the ordered values: 4, 5, 5, 6, 6, 6, 7, 7, 8

Frequencies balance at equal distances from the central peak

Symmetric with one central peak

The mean is 6, and the median is 6

Describe the visible distribution as well as the summaries. The direction of skew names the longer tail, not the end containing most observations

Inspect matching frequencies on both sides

Follow the tail towards larger values

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Which complete description matches the distribution’s tails and peaks?

Read the ordered values: 3, 3, 3, 4, 4, 5, 8, 11

Most observations are at the lower end, with a longer tail towards larger values

Longer tail towards higher values

The mean is 5.125, and the median is 4

Describe the visible distribution as well as the summaries. The direction of skew names the longer tail, not the end containing most observations

Do not name the skew after the tallest stack

Follow the tail towards smaller values

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Which complete description matches the distribution’s tails and peaks?

Read the ordered values: 4, 7, 10, 11, 11, 12, 12, 12

Most observations are at the higher end, with a longer tail towards smaller values

Longer tail towards lower values

The mean is 9.875, and the median is 11

Describe the visible distribution as well as the summaries. The direction of skew names the longer tail, not the end containing most observations

Compare both tails before naming the shape

Two peaks can still be symmetric

Illustrative data record the number of leaves on each seedling

One dot represents one observation, including repeated values

Which complete description matches the distribution’s tails and peaks?

Read the ordered values: 5, 5, 5, 6, 9, 10, 10, 10

There are two equally high separated peaks, with a gap between the groups

Two separated peaks, symmetric about the centre

The mean is 7.5, and the median is 7.5

Describe the visible distribution as well as the summaries. The direction of skew names the longer tail, not the end containing most observations

Describe both the peaks and the balance

The observations are 2, 3, 3, 4 and the movable value t. Move t from 4 to 8 and compare the plot with its summaries

Across these five settings, which statement is correct?

Compare centre and spread separately

Check that both groups record the same variable with the same unit and a compatible collection rule

Read the numerical axes. Two plots may be drawn to different physical scales, so compare their values rather than their widths on the screen

Use mean and median to describe centre and range to describe overall spread

A group with a higher centre can have the same, larger or smaller range

State the group names, summary values and relevant shape features when explaining a comparison

A centre difference without a range difference

Illustrative data show numbers of leaves per seedling

The groups record the same variable and unit. Read each numerical scale rather than comparing the physical width of the plots

Calculate mean, median and range separately for the two plotted groups

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

Group A, ordered observations: 8, 9, 10, 11, 12

There are 5 observations. The total is 50; dividing by 5 gives mean 10

Position 3 gives median 10

Subtract 8 from 12 to obtain range 4

Group B, ordered observations: 11, 12, 13, 14, 15

There are 5 observations. The total is 65; dividing by 5 gives mean 13

Position 3 gives median 13

Subtract 11 from 15 to obtain range 4

Group B has a higher median and the same range than group A

Compare centre and spread separately. A larger centre does not by itself imply a larger spread

Calculate the centres before comparing them

Equal centres with different spreads

Illustrative data show fallen-leaf lengths in centimetres, recorded to the nearest tenth

The groups record the same variable and unit. Read each numerical scale rather than comparing the physical width of the plots

Calculate mean, median and range separately for the two plotted groups

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

Group A, ordered observations: 4.8, 4.9, 5, 5.1, 5.2

There are 5 observations. The total is 25; dividing by 5 gives mean 5

Position 3 gives median 5

Subtract 4.8 from 5.2 to obtain range 0.4

Group B, ordered observations: 4.6, 4.8, 5, 5.2, 5.4

There are 5 observations. The total is 25; dividing by 5 gives mean 5

Position 3 gives median 5

Subtract 4.6 from 5.4 to obtain range 0.8

Group B has the same median and a larger range than group A

Compare centre and spread separately. A larger centre does not by itself imply a larger spread

Use ranges to distinguish matching centres

A higher centre can come with a smaller range

Illustrative data show numbers of leaves per seedling

The groups record the same variable and unit. Read each numerical scale rather than comparing the physical width of the plots

Calculate mean, median and range separately for the two plotted groups

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

Group A, ordered observations: 10, 12, 14, 16, 18

There are 5 observations. The total is 70; dividing by 5 gives mean 14

Position 3 gives median 14

Subtract 10 from 18 to obtain range 8

Group B, ordered observations: 18, 20, 20, 20, 22

There are 5 observations. The total is 100; dividing by 5 gives mean 20

Position 3 gives median 20

Subtract 18 from 22 to obtain range 4

Group B has a higher median and a smaller range than group A

Compare centre and spread separately. A larger centre does not by itself imply a larger spread

Keep the two comparisons independent

Investigate separated observations

A dot well separated from the main group is a possible unusual or extreme observation

Read its actual value and consider the context before deciding whether it needs checking

For a stated comparison, calculate the summaries both with and without only that observation

An extreme affects the mean and range directly. Recalculate the median rather than assuming it stays unchanged

A separated value is not automatically an error. Check the original record before deciding whether a correction is justified

Compare the displayed data with one high value omitted

Illustrative data show numbers of leaves per seedling

The exact value-frequency record accompanies the plot so every observation can be read without estimating crowded positions

For comparison only, omit the single observation at 24 leaves. Retain every other dot

The value is clearly separated from the main group. This does not prove it is an error or justify deleting it from the original record

Calculate the summaries before and after omitting the specified separated observation

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

All observations, ordered observations: 8, 9, 10, 11, 24

There are 5 observations. The total is 62; dividing by 5 gives mean 12.4

Position 3 gives median 10

Subtract 8 from 24 to obtain range 16

After omission, ordered observations: 8, 9, 10, 11

There are 4 observations. The total is 38; dividing by 4 gives mean 9.5

Positions 2 and 3 contain 9 and 10. Average them to obtain median 9.5

Subtract 8 from 11 to obtain range 3

Only 24 is omitted; the remaining record has 4 observations

Mean: 12.4 to 9.5; median: 10 to 9.5; range: 16 to 3

Recalculate all three. The median can change when one observation is removed, even when its change is smaller than the mean’s

Recalculate all summaries after the specified omission

A low separated measurement also changes the comparison

Illustrative data show fallen-leaf lengths in centimetres, recorded to the nearest tenth

The exact value-frequency record accompanies the plot so every observation can be read without estimating crowded positions

For comparison only, omit the single observation at 1 cm. Retain every other dot

The value is clearly separated from the main group. This does not prove it is an error or justify deleting it from the original record

Calculate the summaries before and after omitting the specified separated observation

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

All observations, ordered observations: 1, 3, 3.1, 3.2, 3.3

There are 5 observations. The total is 13.6; dividing by 5 gives mean 2.72

Position 3 gives median 3.1

Subtract 1 from 3.3 to obtain range 2.3

After omission, ordered observations: 3, 3.1, 3.2, 3.3

There are 4 observations. The total is 12.6; dividing by 4 gives mean 3.15

Positions 2 and 3 contain 3.1 and 3.2. Average them to obtain median 3.15

Subtract 3 from 3.3 to obtain range 0.3

Only 1 is omitted; the remaining record has 4 observations

Mean: 2.72 to 3.15; median: 3.1 to 3.15; range: 2.3 to 0.3

Recalculate all three. The median can change when one observation is removed, even when its change is smaller than the mean’s

Read the low observation and retain every other dot

Group A has mean and median 14 leaves and range 8 leaves. Group B has mean and median 20 leaves and range 4 leaves. Which description is supported?

Use software without losing the original observations

Keep one recorded value per row in the original data column

Sort a copy rather than deleting repeated entries. A frequency table should count every occurrence

For a dot plot, use the numerical values as horizontal positions and plot one point for each occurrence, stacked at heights one, two and so on

For a stem plot, use the stated precision to split each value into stem and leaf, then order leaves within each row

After software creates a display, check the labels, unit, key, gaps and total number of observations against the source record

Construct and explain two displays of one record

Use this illustrative record of seed counts: 21, 24, 24, 25, 30, 32, 32, 35

Draw a labelled dot plot and an ordered stem-and-leaf plot with a key. Use software if available, keeping every repeated observation

Check that both displays contain eight observations and calculate their mean, median, mode and range

Explain which features of the record are easier to see in each display. Describe any clusters or gaps without declaring an error from appearance alone

Then compare this record with 18, 22, 24, 24, 30, 32, 36, 38 using the same units and numerical summaries

The independently drawn layouts, software actions and written explanations are not automatically checked

A display and its summaries describe the same observations

Retain each observation, including repeats, and use an accurate scale or stem-and-leaf key

Check the number of dots or leaves against the original record, including zero-frequency gaps and empty stems

Recover the observations before calculating mean, median, mode and range

Describe tails, peaks, symmetry, clusters and gaps as well as numerical centre and spread

Compare groups using the same variable and units, and investigate separated values without deleting them automatically