Find the mean and the median, and choose between them

Prices of the six bikes

\(\displaystyle \begin{gathered}\$180,\ \$220,\ \$250,\\\$260,\ \$300,\ \$1{,}150\end{gathered}\)

Six second-hand bikes are for sale. Their prices are $180, $220, $250, $260, $300 and $1,150. What is the median price of the six bikes?

Move the stretch from 0 to 14. Seven delivery times begin symmetrically around 28 minutes. Compare the mean and median as the three longest times increase

What happened as the three longest deliveries stretched out?

Values, positions and frequencies

The mean is the total divided by the number of observations. The median is a central value after sorting, not a position number. A frequency tells how often a value occurs

1. 1Odd count Take the value at the central position
2. 2Even count Average the two central values
3. 3Frequency data Count every observation, including repeated values and zeros
4. 4Purpose A typical central value and a total-based comparison can require different summaries

Recorded values

\(\displaystyle 12,3,8,5,9\)

The recorded values are 12, 3, 8, 5, 9. Find the median

Sorted values

\(\displaystyle 2,4,7,11\)

The sorted values are 2, 4, 7, 11. Find the median

Rainfall readings, in recorded order, are 4, 0, 6, 2, 43, 1, 5, 3 mm. Find the mean

Find the median rainfall from the same table

For eight days, the rainfall readings were 0, 1, 2, 3, 4, 5, 6 and 43 mm. Which summary better describes a typical day without being pulled far upwards by the 43 mm day?

Days

\(\displaystyle 8\)

Mean rainfall (mm)

\(\displaystyle 8\)

Using the mean 8 mm over eight days, recover the total rainfall in millimetres

For eight rainfall days, the mean was 8 mm and the median was 3.5 mm. Which summary reconstructs the recorded total?

A

\(\displaystyle 0,\ 0,\ 10,\ 10\)

B

\(\displaystyle 4,\ 5,\ 5,\ 6\)

Both lists have mean 5 and median 5. Must their distributions be alike?

The ordered list is 1, 2, 3, 4. Remove 4. Does the median stay unchanged?

Three recorded temperatures are 0, 10 and 10 °C. Where is the mean relative to the median?

Group A has 5 observations with mean 6 min; group B has 15 with mean 10 min. Predict the combined mean

Change A’s count from 5 to 15 and then 30 while both group means remain fixed. Watch how the totals and the combined mean respond

Why is the combined mean closer to 6 when A has 30 observations?

Group A

\(\displaystyle n_A=5,\quad\bar{x}_A=6\)

Group B

\(\displaystyle n_B=15,\quad\bar{x}_B=10\)

For 5 observations averaging 6 min and 15 averaging 10 min, calculate the combined mean in minutes

Recover each total, add them, then divide by the combined count

Can group medians and group counts always reconstruct the combined mean?

Choose a summary for the question

A total-based planning question can need a mean even when data are skewed. A resistant description of a typical observation may favour the median. Report the sample, units, shape, spread and collection limits as well

1. 1Do not discard inconvenient values Investigate accuracy and relevance before excluding a record. Report any justified exclusion
2. 2Independent explanation Compare lists 1, 2, 3 and 1, 2, 30. Write their medians and explain which mean would recover the correct total. Then describe what their identical medians conceal. Review your writing yourself or with a teacher
3. 3Complete a comparison record Use sample A: 1, 2, 3 and sample B: 1, 2, 30. Record each count, total, mean and median. Explain what their equal medians conceal and which statistic reconstructs each total. These are abstract numerical records; no measurement units have been supplied
4. 4Choose and use technology Choose statistical functions or spreadsheet formulas to calculate both summaries. Explain why your functions suit the raw lists, and record all inputs, ranges, formulas or settings and outputs. Include the count and sum so your mean can be checked
5. 5Verify independently Sort and add each list by hand. Identify its middle record, check mean × count against its total, and test the claim that median × count always reconstructs the total. Explain why this does not make the median invalid for describing a typical observation

Model response: a summary must answer the question

Both samples have three records and median 2, but their totals and means differ. Sample A totals 6 and has mean 2; sample B totals 33 and has mean 11. A median alone does not determine a total

1. 1One reproducible tool record Enter 1, 2, 3 in A2:A4 and 1, 2, 30 in B2:B4. For each column use COUNT for sample size, SUM for total, AVERAGE for mean and MEDIAN for the middle value. For example, AVERAGE(B2:B4) returns 11 and MEDIAN(B2:B4) returns 2. Use numerical entries and check the exact three-cell ranges; do not include headings or an added Total row
2. 2Independent arithmetic For A, adding 1, 2 and 3 gives 6; dividing by 3 gives a mean of 2. Its middle ordered record is 2. For B, adding 1, 2 and 30 gives 33; dividing by 3 gives a mean of 11. Its middle ordered record is also 2. Multiplying each mean by its count gives 6 and 33 respectively. Multiplying each median by its count gives 6 for both, failing to recover B’s total
3. 3What the identical medians conceal Replacing A’s largest record 3 with 30 leaves the middle record at 2, but increases the total by 27 and the mean by 9. The ranges are 2 and 29. The median remains useful as a resistant description of the central record; the mean is required to reconstruct the total. The mean 11 need not be an observed record
4. 4Use a counterexample carefully B disproves the claim that median × count always equals the total. A happens to satisfy it because its mean and median agree. One success cannot establish the universal claim. The unchanged median here also does not prove that every possible change or removal leaves every median unchanged
5. 5Self-review Check both sample sizes and totals, the sorted middle records, exact means, the stated purpose, the counterexample, complete tool inputs/functions and the independent arithmetic. Other tools are valid if they reproduce these calculations. Written reasoning and tool operation remain self or teacher reviewed

For the records 1, 2, 30, which statement correctly distinguishes the two summaries?

A sample has mean 10 minutes. Which claim is justified by that fact alone?