Plan, analyse and report a statistical investigation

A student records one leaf count for each of twelve seedlings. What does one data entry represent?

Start with a question that anticipates variation

A statistical question asks about a group of observations whose values may vary

For example: how do leaf counts vary among twelve specified seedlings on one day? Each seedling supplies one discrete count

A name label is not a numerical observation, and the height of one specified object alone is not a distribution

State what group, place and period the question concerns before choosing the data to collect

Choose a question about variation

Distinguish direct records from existing records

Primary data are collected directly for the investigation, such as counting leaves on specified seedlings

Secondary data already collected by another source can be analysed for a new question

For secondary data, record the original organisation, table, dates, units, definitions and access date

Primary does not mean automatically reliable, and secondary does not mean automatically unreliable. Check the collection process and suitability

Identify the source relationship

Define counted and measured variables separately

For twelve seedlings, count attached individual leaves using one stated rule. One seedling supplies one count; the possible counts are discrete

For fallen leaves, measure base-to-tip length in centimetres to the nearest tenth. One leaf supplies one measurement of an underlying continuous variable

Keep these two records separate. A seedling leaf count and a leaf length are not values of the same variable

Selecting exactly eight leaves and then counting those selected leaves gives a fixed total of eight, not eight variable count observations

Distinguish the variable from its recording precision

Plan a consistent collection before seeing the result

Choose the observation group and selection rule without deliberately choosing only values that support a preferred result

For two areas, record their names, date, sample sizes and the same leaf-selection rule

Measure every leaf from the same defined base to tip, using the same unit and precision

Keep original observations, repeats and procedure notes. A small convenience collection describes those records, not every leaf in either area

Keep the comparison procedure compatible

Put the investigation steps in a defensible order

Check a record without inventing observations

A repeated value is not automatically a duplicate error; different observations can genuinely match

A missing or unreadable measurement is not a measured zero

Check the original note or repeat the measurement if appropriate and document what was done

If recovery is impossible, mark the entry missing, state the valid observation count and acknowledge the limitation

Handle an unreadable measurement honestly

Analyse an illustrative primary-data record

The supplied leaf records on the next screens are illustrative, not measurements collected by this learner

Order the complete valid record to locate the median and modal values

Add every observation and divide by its count for the mean. Subtract minimum from maximum for range

Use a labelled dot plot or ordered stem-and-leaf display to describe the distribution as well as calculating its summaries

Summarise five illustrative seedling counts

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, 7

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, 7

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

7 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 7 and the smallest value is 3

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

Calculate a second count record

Use the same definitions for measurements

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, 4.3, 4.3, 4.7, 5.1

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: 4.1, 4.3, 4.3, 4.7, 5.1

There are 5 values, so use position 3

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

Count every occurrence, including repeats

4.1 occurs 1 time

4.3 occurs 2 times

4.7 occurs 1 time

5.1 occurs 1 time

The greatest frequency is 2, so the mode is 4.3

A mode is a data value, not its frequency

Use the smallest and largest observed values

The largest value is 5.1 and the smallest value is 4.1

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

Keep every decimal measurement

Let the display reveal features that one average omits

A dot plot keeps one dot per numerical observation; an ordered stem-and-leaf display keeps one leaf per observation and needs a key

Both must preserve every original value and repeat. Changing representation does not change the mean or median

Look for the main concentration, gaps, peaks, approximate symmetry and values that extend one side

For a small record, describe visible features cautiously rather than claiming that its shape proves a cause or a whole-population pattern

Check two displays against the same record

This illustrative record has times of 4, 5, 5 and 6 seconds, plus a fifth value. Move the fifth value between 6, 13 and 20 seconds to see how a separated value affects the display

Which statement holds across these three settings?

Choose a centre for a stated reporting purpose

Choose the quantity that answers the question

A total answers how much there is altogether; a mean answers the total per observation

When groups contain different numbers of observations, their totals alone do not compare their means

For four leaves totalling 20 cm, mean length is 20 divided by 4 = 5 cm. For eight leaves totalling 32 cm, it is 32 divided by 8 = 4 cm

The second total is greater, but its mean is smaller. Both are legitimate quantities for different questions; report the group sizes

Use each group’s own observation count

Investigate an identified published numerical record

The next questions reuse published Australian Bureau of Statistics daily marriage counts for 1 to 14 January 2023

One observation is a date’s published count, not one marriage. There are seven dates in each displayed week and fourteen in the combined record

This is a dated published snapshot, including registrations received in 2024. It is not a claim that every eventual registration is present

A daily time plot answers how values change through dates. A dot plot instead shows a distribution without date order. Here its labelled nearest-hundred counts give a readable overview; retain the exact dated table for calculations

Analyse the first seven published dates

Published secondary data: Australian marriages by date of occurrence in January 2023

One observation is one date’s published marriage count, not one individual marriage

Source: Australian Bureau of Statistics, Marriages and Divorces, Australia, 2023; released 16 August 2024

https://www.abs.gov.au/statistics/people/people-and-communities/marriages-and-divorces-australia/2023

Accessed 25 October 2026; Commonwealth of Australia, CC BY 4.0; table and plot redrawn from the published snapshot

These published occurrence counts include registrations received in 2024. They are not a claim of complete eventual counts

For a readable overview only, the dot plot rounds each daily count to the nearest hundred. Keep the original table for every exact calculation

The dot plot shows the distribution of rounded daily counts, not their time order. Its grouped positions are approximate; different exact counts can share a rounded position. Calculate all summaries, including modes, from the exact table

Calculate each requested exact summary from the published record

Keep exact integers, terminating decimals or simplified fractions without rounding

Supplied dates: order the 7 daily counts as 61, 70, 117, 146, 202, 366, 818

Their total is 1780. Divide by 7 dates to obtain mean daily count \(\frac{1780}{7}\)

Position 4 gives median 146

Maximum 818 minus minimum 61 gives range 757

All supplied counts are different, so there is no mode under the convention used here

The large upper values extend the distribution above the main group; a mean alone does not describe that shape

The conclusions describe only the dates supplied in this published snapshot, not the whole year or a cause of the pattern

Analyse the next seven dates independently

Compare two specified seven-day records

Published secondary data: Australian marriages by date of occurrence in January 2023

One observation is one date’s published marriage count, not one individual marriage

Source: Australian Bureau of Statistics, Marriages and Divorces, Australia, 2023; released 16 August 2024

https://www.abs.gov.au/statistics/people/people-and-communities/marriages-and-divorces-australia/2023

Accessed 25 October 2026; Commonwealth of Australia, CC BY 4.0; table and plot redrawn from the published snapshot

These published occurrence counts include registrations received in 2024. They are not a claim of complete eventual counts

For a readable overview only, the dot plot rounds each daily count to the nearest hundred. Keep the original table for every exact calculation

The dot plot shows the distribution of rounded daily counts, not their time order. Its grouped positions are approximate; different exact counts can share a rounded position. Calculate all summaries, including modes, from the exact table

Calculate each requested exact summary from the published record

Keep exact integers, terminating decimals or simplified fractions without rounding

1 to 7 January: order the 7 daily counts as 61, 70, 117, 146, 202, 366, 818

Their total is 1780. Divide by 7 dates to obtain mean daily count \(\frac{1780}{7}\)

Position 4 gives median 146

Maximum 818 minus minimum 61 gives range 757

8 to 14 January: order the 7 daily counts as 98, 136, 138, 161, 219, 233, 943

Their total is 1928. Divide by 7 dates to obtain mean daily count \(\frac{1928}{7}\)

Position 4 gives median 161

Maximum 943 minus minimum 98 gives range 845

The second displayed week has a higher mean, a higher median and a larger range

These are two particular consecutive weeks, not random samples establishing a pattern for every week

The conclusions describe only the dates supplied in this published snapshot, not the whole year or a cause of the pattern

Calculate summaries for all fourteen dates

Ordered Australian published daily marriage counts for 1 to 14 January 2023 are 61, 70, 98, 117, 136, 138, 146, 161, 202, 219, 233, 366, 818 and 943. Their mean is 1854 divided by 7, in marriages per date; the median is 153.5 and the range is 882. Which report is supported?

Compare sensitivity without declaring an error

A separated large observation can affect centre and spread, but separation alone does not establish a recording error

For a stated comparison, keep the original record and calculate a second summary with only the specified value omitted

Recalculate the observation count and the median positions as well as mean and range

Label the modified record clearly. A sensitivity comparison is not a reason to silently delete the original value

Recalculate after one explicitly specified omission

Published secondary data: Australian marriages by date of occurrence in January 2023

One observation is one date’s published marriage count, not one individual marriage

Source: Australian Bureau of Statistics, Marriages and Divorces, Australia, 2023; released 16 August 2024

https://www.abs.gov.au/statistics/people/people-and-communities/marriages-and-divorces-australia/2023

Accessed 25 October 2026; Commonwealth of Australia, CC BY 4.0; table and plot redrawn from the published snapshot

These published occurrence counts include registrations received in 2024. They are not a claim of complete eventual counts

For a readable overview only, the dot plot rounds each daily count to the nearest hundred. Keep the original table for every exact calculation

The dot plot shows the distribution of rounded daily counts, not their time order. Its grouped positions are approximate; different exact counts can share a rounded position. Calculate all summaries, including modes, from the exact table

For a comparison only, omit the single highest daily count, 818. This is not a decision to delete a valid original observation

Calculate each requested exact summary from the published record

Keep exact integers, terminating decimals or simplified fractions without rounding

All supplied dates: order the 7 daily counts as 61, 70, 117, 146, 202, 366, 818

Their total is 1780. Divide by 7 dates to obtain mean daily count \(\frac{1780}{7}\)

Position 4 gives median 146

Maximum 818 minus minimum 61 gives range 757

After the specified omission: order the 6 daily counts as 61, 70, 117, 146, 202, 366

Their total is 962. Divide by 6 dates to obtain mean daily count \(\frac{481}{3}\)

Middle positions 3 and 4 contain 117 and 146; their average is 131.5

Maximum 366 minus minimum 61 gives range 305

Omit only the count 818; the remaining record contains 6 dates

Recalculate the middle positions as well as the mean and range. Investigate the original record before deciding whether a value is erroneous

The conclusions describe only the dates supplied in this published snapshot, not the whole year or a cause of the pattern

Keep the original and modified records distinct

Keep a conclusion within its evidence

State the group, place, dates and collection method represented in a comparison

A higher median does not prove every individual value is greater

Two particular consecutive weeks are not random samples that establish what happens in every week of a year

An observational difference does not isolate a cause when other relevant factors differ or were not controlled

Separate a sample summary from an unlimited claim

Do not turn a difference into proof of cause

Write a report that can be checked

State the question and describe the observation units, numerical variables, data source and collection procedure

Give sample sizes, labelled displays and relevant centre and spread calculations

Describe visible distribution shape using the same record, then answer the question with numerical evidence

Discuss missing values, unusual observations and limits on generalising or explaining causes

For online data, cite the original source and definitions rather than relying only on a copied graphic

Link the distribution and summaries to the observed group

Check a published claim before relying on it

Collect only the information the question needs

A numerical investigation should not add identifying personal details without a relevant need

An optional timed classroom activity can use an anonymous trial code, elapsed seconds and procedure notes

Explain the activity and allow a non-personal alternative such as measuring leaves or counting counters

Keep the collection appropriate to the classroom and avoid reporting named individual results

Keep a useful record without unnecessary identifiers

Plan and conduct a complete investigation

Written and practical investigation task

Choose a question about variation in leaf counts among twelve available seedlings on a stated day. Count attached leaves for each seedling using one documented rule; each seedling gives one discrete observation

Separately, compare lengths of eight fallen leaves from each of two specified areas. Define selection, base-to-tip measurement, centimetres and nearest-tenth precision before collecting; each leaf gives one continuous measurement

Use safe, permitted collection locations and retain the original labelled records, repeats and any missing-value notes. Do not combine lengths and counts into one distribution

Construct suitable displays by hand or with software. Check their observation counts and keys or scales, calculate summaries, and describe visible shape

For secondary-data practice, use the identified ABS record from this topic. Keep the published source, dates, units and snapshot limitation with your analysis

Write a report answering each question using numerical evidence, shape and limitations. Explain why your collection does not automatically represent every plant, area, season or date

Actual data collection, independent displays, software actions and the full written report are not automatically checked; the topic checks only supplied calculations and selected investigation decisions

Keep the question, record and claim connected

Define the observation unit and numerical variable, then plan a consistent and appropriate collection

Distinguish primary records from identified secondary sources and retain provenance and original observations

Check missing values and unusual records without inventing or silently deleting data

Use displays, shape, centre and spread together; choose quantities according to the question

Report numerical evidence with sample sizes, units, dates and limits. Selected checks do not certify independently conducting an entire investigation