Describe the shape of a distribution

The histogram shows puzzle-completion times. Which way does its long thin tail run?

Move tilt from 1 to 4. At 2.5 the data are symmetric. Watch the actual dots and centre markers as the longer tail changes sides

In this constructed example, what happens as tilt moves from 1 to 4?

Describe what the display shows

Consider the overall pattern: clusters, peaks, tails, gaps and isolated observations. Positive and negative skew describe tail direction, not the sign of the values

1. 1Positive The longer thin tail extends towards larger values
2. 2Negative The longer thin tail extends towards smaller values
3. 3Symmetry The two sides roughly mirror one another about the centre
4. 4Evidence Use the actual plot and its scale. A single summary cannot determine the whole shape

What shape does this histogram suggest?

What shape does this histogram show?

All values in a data set are positive. Must its distribution be positively skewed?

This distribution has two peaks. Can it also be symmetric?

Days absent

\(\displaystyle \begin{gathered}0,1,1,2,2,2,\\3,3,4,6,9\end{gathered}\)

For these 11 absence values, what is the median number of days?

For the same 11 values, what is the mean?

Which is less affected by the unusually high value of 9 days: the mean or the median?

A report needs a total-based average that accounts for every observation. Which statistic matches that purpose?

Describe the main body of this plot, setting aside the isolated value 30

Group B is Group A shifted 20 units higher. What changes?

These groups share median 30. How do their spreads compare?

Two plots fill the same physical width. Can you conclude that their ranges are equal?

What can equal mean and median tell you on their own?

Write a complete distribution comparison

Two small invented samples record waiting times in minutes. Make a dot plot for each on the same numerical scale, then write a short comparison. Describe shape, centre, spread and unusual features with evidence. State what these samples can and cannot establish

1. 1Sample A (minutes) 1, 2, 2, 3, 3, 3, 4, 4, 5, 9
2. 2Sample B (minutes) 3, 4, 4, 5, 5, 5, 6, 6, 7
3. 3Compare like features Count each sample separately; compare medians and full ranges with units. Identify the main body and any isolated observation separately from the whole distribution. Mention overlap. Do not infer a bell-shaped population simply from a symmetric sample
4. 4Investigate the unusual record Explain what you would check before deciding whether to remove the 9-minute observation. Record how removing it would change the count, median, range and shape; do not silently remove it from your primary comparison
5. 5Complete before continuing The drawings and writing are not automatically marked; use self or teacher review. Automatic practice checks its named components only

Model response: compare the recorded samples

A has ten recorded waits and B has nine. B has the higher median, 5 min versus 3 min, but the smaller full range, 4 min versus 8 min. These are comparisons of these samples, not claims about every wait or the cause of a difference

1. 1Complete plots Use a common horizontal scale from 0 to 10 minutes with unit spacing. For A, the frequencies at 1, 2, 3, 4, 5 and 9 are 1, 2, 3, 2, 1 and 1. Keep the empty positions 6, 7 and 8. For B, frequencies at 3, 4, 5, 6 and 7 are 1, 2, 3, 2 and 1. Use one dot per record and name both groups
2. 2Shape and unusual features B is symmetric about 5 min. A’s main body from 1 to 5 min is symmetric about 3 min, with one isolated high observation at 9 min. A as a whole is not symmetric; report the high-side extension as well as its symmetric main body. Symmetry alone does not establish a normal population model
3. 3Numerical evidence A’s fifth and sixth ordered values are both 3, giving median 3 min. Its range is 9−1=8 min. B’s fifth value is 5, giving median 5 min, and its range is 7−3=4 min. Both samples contain 3, 4 and 5 min, so the higher median does not mean every B wait exceeds every A wait
4. 4Investigate before changing data Check the original record, units, collection process and whether this was a genuine long wait. The plot alone does not prove an error. Removing 9 would leave A with nine observations, median 3 min and range 5−1=4 min; the remaining plot would be symmetric. Report any justified correction and its effect. Keep 9 unless there is a stated defensible reason to exclude it; no formal outlier rule is being applied here
5. 5Limits and self-review Check complete plots, numerical scales, both sample sizes, like-for-like statistics with units, the isolated record, overlap and the stated scope. Small samples with unspecified selection methods do not establish population differences or causation. This model supports review of your work; it does not certify that the independent work was completed

Which description of sample A preserves all ten observations?