Find the range, the quartiles and the IQR

Seven walking times, in minutes, are 12, 7, 25, 9, 18, 4 and 15. What is their range?

The seven observations are 6, 9, 12, 15, 19 and the two slider values. Move one slider at a time. Watch the actual observations as well as the spread readouts

In this example, what changes when only the largest value moves?

Two ways to describe spread

Compare the full span with the middle-half spread. State a quartile convention and retain every repeated observation

1. 1Range Subtract the minimum from the maximum. It uses only the extremes
2. 2Quartile convention Here, order every observation, retaining repeats. For odd n, omit only the middle observation from the two halves; for even n, split without omitting any observation
3. 3Quartiles Q1 is the median of the lower half. Q3 is the median of the upper half. Other software may use a different convention, so state your method
4. 4IQR Subtract Q1 from Q3. It is a spread, not a position or the number of observations

Ordered counts

\(\displaystyle 4,\ 7,\ 9,\ 12,\ 15,\ 18,\ 25\)

What is the median of these seven ordered observations?

Order the observations and use the stated median-of-halves convention

Ordered counts

\(\displaystyle 4,\ 7,\ 9,\ 12,\ 15,\ 18,\ 25\)

Omit the middle observation 12. What is Q1?

Order the observations and use the stated median-of-halves convention

Ordered counts

\(\displaystyle 4,\ 7,\ 9,\ 12,\ 15,\ 18,\ 25\)

Using the same convention, what is Q3?

Order the observations and use the stated median-of-halves convention

Ordered counts

\(\displaystyle 4,\ 7,\ 9,\ 12,\ 15,\ 18,\ 25\)

What is the IQR?

Order the observations and use the stated median-of-halves convention

An even number of observations

Order these eight times. The median lies between the fourth and fifth observations. Each half contains four observations

1. 1Lower half 7, 9, 12, 14. Average its two middle observations to find Q1.
2. 2Upper half 16, 18, 20, 22. Average its two middle observations to find Q3.
3. 3Keep every observation Do not insert the median into the list or remove either central observation

Recorded times (minutes)

\(\displaystyle \begin{gathered}18,\ 9,\ 22,\ 12\\16,\ 7,\ 20,\ 14\end{gathered}\)

Find Q1 for the eight recorded times

Order the observations and use the stated median-of-halves convention

Recorded times (minutes)

\(\displaystyle \begin{gathered}18,\ 9,\ 22,\ 12\\16,\ 7,\ 20,\ 14\end{gathered}\)

Find Q3 for the same eight times

Order the observations and use the stated median-of-halves convention

Recorded times (minutes)

\(\displaystyle \begin{gathered}18,\ 9,\ 22,\ 12\\16,\ 7,\ 20,\ 14\end{gathered}\)

Find the IQR of the eight times

Order the observations and use the stated median-of-halves convention

Recorded lengths (metres)

\(\displaystyle \begin{gathered}1.2,\ 1.4,\ 1.4,\ 1.6\\1.8,\ 2.0,\ 2.2,\ 2.6\end{gathered}\)

Use the median-of-halves convention. What is Q1 for these eight recorded lengths?

Recorded lengths (metres)

\(\displaystyle \begin{gathered}1.2,\ 1.4,\ 1.4,\ 1.6\\1.8,\ 2.0,\ 2.2,\ 2.6\end{gathered}\)

What is Q3 for these lengths?

Recorded lengths (metres)

\(\displaystyle \begin{gathered}1.2,\ 1.4,\ 1.4,\ 1.6\\1.8,\ 2.0,\ 2.2,\ 2.6\end{gathered}\)

What is the IQR of these lengths?

Five observations are 4, 4, 4, 4 and 4. Which statement is correct?

For the ordered counts 4, 7, 9, 12, 15, 18, 25, a student calculates Q1 as 4 + (25 − 4) ÷ 4. What is the mistake?

Specify the measure when comparing

These are hypothetical travel-time samples in minutes. Both have median 6. Compare their full spans and their middle halves separately

1. 1Sample A 0, 2, 4, 6, 8, 10, 12. Range 12; Q1 2; Q3 10; IQR 8.
2. 2Sample B 0, 4, 5, 6, 7, 8, 30. Range 30; Q1 4; Q3 8; IQR 4.
3. 3Interpretation B has the larger overall span, but A has the more spread-out middle half. Neither summary describes every feature

Ranges (minutes)

\(\displaystyle A:12,\quad B:30\)

Which sample has the larger full span?

IQRs (minutes)

\(\displaystyle A:8,\quad B:4\)

Which sample has the more spread-out middle half?

Is it always true that adding or moving any unusual observation leaves the IQR unchanged?

Original seven values

\(\displaystyle 0,\ 2,\ 3,\ 4,\ 5,\ 6,\ 12\)

Additional observation

\(\displaystyle 30\)

The original seven values have Q1 = 2 and Q3 = 6. Add 30 as an eighth observation. What is the new Q3?

Original seven values

\(\displaystyle 0,\ 2,\ 3,\ 4,\ 5,\ 6,\ 12\)

Additional observation

\(\displaystyle 30\)

After adding 30, what is the new IQR?

The dot plot shows eight values. Find their IQR using the stated convention

Order the observations and use the stated median-of-halves convention

Use the stem-plot key to find the range

Use the key to identify the smallest and largest values, then subtract the smallest from the largest

Reading spread from a boxplot

An ordinary boxplot summarises minimum, Q1, median, Q3 and maximum. Modified boxplots show outliers separately and use non-outlier whisker endpoints; those follow later in this section

1. 1Range For this ordinary boxplot, read the minimum and maximum and subtract
2. 2IQR Read the two box edges and subtract. The median line inside the box is not an edge
3. 3Limits A boxplot does not show every observation or the sample size. More construction and interpretation follows in this section and 5F

The box edges are at 5 and 11. Find the IQR

Read the two edges of the box and subtract Q1 from Q3.

A has observations 0, 2, 4, 6, 8, 10, 12. B has 0, a value of 2 increased by k, 6, 6, 6, a value of 10 decreased by k, and 12. Change k from 0 to 2 and 4; compare both boxes on the same scale

What changes as the inner observations in B approach 6?

Write a supported comparison

Use samples A and B below. Draw a dot plot for each on identical scales and write two sentences comparing spread. State the measure and units

1. 1Data A: 0, 2, 4, 6, 8, 10, 12 minutes. B: 0, 4, 5, 6, 7, 8, 30 minutes
2. 2Check your evidence B has the larger range (30 versus 12 minutes); A has the larger IQR (8 versus 4 minutes). Both medians are 6 minutes
3. 3Review the conclusion Mention the isolated high value in B. Do not claim a cause or that the samples represent all travellers. Drawings and written explanations need self or teacher review
4. 4Choose and verify a tool Use a statistical tool or spreadsheet to calculate both ranges and quartiles. State the median-of-halves convention, excluding the central observation for odd counts. Explain your chosen functionality and record inputs, settings and results. Check it manually; a default percentile algorithm may use a different convention
5. 5Complete before continuing Save your plots, comparison and tool record before opening the model. The complete drawing, writing and external-tool execution are not automatically marked

Model response: specify the measure and convention

B has the larger full span, 30 versus 12 minutes, while A has the larger middle-half spread, IQR 8 versus 4 minutes. Both medians are 6 minutes. B’s isolated high record affects its range; it does not justify silently removing that record or claiming a cause

1. 1Check each half A sorts as 0, 2, 4, 6, 8, 10, 12. Exclude median 6; lower 0, 2, 4 gives Q1=2 and upper 8, 10, 12 gives Q3=10. B sorts as 0, 4, 5, 6, 7, 8, 30; its halves give Q1=4 and Q3=8. Subtract extremes for range and Q1 from Q3 for IQR
2. 2One reproducible workflow Put each sorted seven-record sample in its own column, rows 2–8. Use MAX(range)−MIN(range) for full span and MEDIAN(range) for centre. Under the stated convention, Q1 is MEDIAN(rows 2–4) and Q3 is MEDIAN(rows 6–8); subtract to obtain IQR. Sorting and taking the medians of explicit halves implements the required definition without assuming that a function named QUARTILE uses it
3. 3Compare conventions rather than blaming arithmetic An inclusive linear percentile convention using a rank found by multiplying p by one less than n, then adding 1, would use positions 2.5 and 5.5 when n=7. It gives A quartiles 3 and 9, and B quartiles 4.5 and 7.5. Those differ from the required median-of-halves results because the definitions differ. Record and use the requested convention consistently; do not average outputs from different conventions
4. 4Complete drawing and report review Use a common 0–30-minute numerical axis for both dot plots, retaining every observation and the large gap before B’s 30. Check seven dots per sample. State both numerical comparisons with units and qualify them as descriptions of these samples. Range and IQR answer different questions; neither number describes the entire distribution

A tool reports A’s quartiles as 3 and 9 for 0, 2, 4, 6, 8, 10, 12. The task requires medians of halves with the central observation excluded. What should you do?