L-DS-3 · Descriptive statistics
Central Tendency
Mean, median, mode—and the honest choice.
One dataset can support more than one correct centre.
Opening question · constructed data
Can a correct average still mislead?
Seven fictional monthly rents near campus. Every value is visible; nothing here is a current market claim.
Six ordinary listings
“Average rent near campus: $1,650”
Anonymous check
Which explanation fits both facts?
Six fictional listings lie between $820 and $990. A seventh penthouse rents for $6,100. Their arithmetic mean is $1,650.
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Both can be right; the penthouse pulls the mean far above most listings
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The headline must be false because six listings are below $1,000
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The ordinary listings must be irrelevant because the mean uses all seven values
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No centre can be calculated when one value is extreme
Lesson map
Five concepts build one defensible centre.
Learn each concept before comparing and choosing among them.
- C1 Sample mean Compute the arithmetic average.
- C2 Median Sort and locate the middle.
- C3 Outlier sensitivity Compare how the mean and median respond.
- C4 Mode Identify what appears most frequently.
- C5 Choose a centre Use data type, shape, and outliers.
C1 · Sample mean
The sample mean is the arithmetic average.
The sample mean, read “x-bar,” is the arithmetic average of n observations.
\(\bar{x}=\frac{\sum_{i=1}^n x_i}{n}=\frac{x_1+x_2+\cdots+x_n}{n}\)
\(x_i\) is the \(i\)-th observation, and \(n\) is the sample size.
The mean can lie between observations; it does not need to be one observed value.
C1 · Constructed rent case
First fact-check the ordinary listings.
Monthly rent in dollars
\(\sum x_i=820+860+900+920+960+990=5{,}450\)
\(\bar{x}=\frac{5{,}450}{6}\approx908.33\)
About $908 lies inside the ordinary-listing range.
C1 · Flex visual
Where will the beam balance?
Predict first, then move the fulcrum. The mean appears when the beam balances.
C2 · Median
The median divides sorted data into two halves.
Its location depends on the number of observations.
The median is the value that divides a sorted dataset into two equal halves.
Smallest to largest.
Find the middle position or pair.
Odd: one middle. Even: average two middles.
An even-sample median may not be an observed value.
C2 · Flex visual
Sorting changes the question from order to position.
Step through raw order, sorted order, middle position, and median.
C2 · Odd sample
Seven rents have one middle.
The values are already sorted.
\(\text{middle position}=\frac{7+1}{2}=4\)
\(\text{median}=\$920\)
C2 · Even sample
Eight rents have two middle values.
Add one $1,040 listing and keep the values sorted.
\(\text{middle positions}=4\text{ and }5\)
\(\text{median}=\frac{920+960}{2}=\$940\)
Anonymous check
What is the median?
The recorded values are 11, 4, 6, 9, 2, 8, 15, 12.
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8
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8.5
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9
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9.5
Anonymous check
Which centre changed more when the penthouse was added?
Before the penthouse is added, the six ordinary rents have mean ≈ $908 and median = $910. Now add one $6,100 rent.
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The mean and median changed by about the same amount
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The mean changed much more than the median
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The median changed much more than the mean
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Neither measure changed
C3 · Before and after
One value moves the two centres differently.
The same $6,100 value enters both calculations.
| Centre | Six ordinary rents | + $6,100 penthouse | Change |
|---|---|---|---|
| Mean | $908.33 | $1,650 | +$741.67 |
| Median | $910 | $920 | +$10 |
The mean follows magnitude; the median follows ordered position.
C3 · Flex visual
The mean follows magnitude; the median follows position.
Move one extreme home price. Watch the two markers—not just the final numbers.
The mean uses every magnitude. The median depends mainly on ordered position.
C3 · Interpretation
Sensitivity is useful—and risky.
A property can help in one distribution and mislead in another.
Useful when all values fairly describe one centre; vulnerable to extreme tails.
Resists tail magnitude; can still change when observations cross the middle.
Resistance is not immunity. Sensitivity is not automatically a defect.
C4 · Mode
Mode asks what appears most.
Frequency—not arithmetic—creates this centre.
The mode is the value—or values—that appears with the highest frequency.
2, 3, 3, 4 → mode 3
2, 2, 3, 3, 4 → modes 2 and 3
2, 3, 4 → course convention
Nominal data permit a mode; ordinal data may also permit a median.
C4 · Flex visual
A tally makes the mode visible.
Predict from the raw values, then build the frequency chart.
Anonymous check
Which mode conclusion is correct?
Hoodie-size sales: S = 3, M = 9, L = 9, XL = 4.
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M only, because it appears first among the tied leaders
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L only, because it is the larger size
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M and L; the distribution is bimodal
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No mode, because the highest frequency is tied
C5 · First decision
Type comes before arithmetic.
A licensed calculation depends on what the values mean.
Most frequent category
Order is meaningful
Then compare centres
C5 · Distribution shape
Skew follows the longer tail.
A skewed distribution has one longer tail. Follow that tail to name the direction.
Anonymous check
Which description matches the shape?
A quantitative histogram peaks on the left. Its bars taper through several small frequencies toward larger values on the right.
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Right-skewed
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Left-skewed
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Approximately symmetric
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Bimodal
Right-skewed. The longer tail points toward larger values.
Anonymous check
Which primary centre best describes a typical student?
Monthly earnings are quantitative. Most students earn $1,500–$3,500, while a small number earn up to $15,000. The histogram has a long right tail.
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Mean, because every quantitative dataset must use all magnitudes
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Median, because the right tail can pull the mean above most students
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Mode, because a histogram always has a tallest bar
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No centre, because skew prevents any summary
C5 · Synthesis
Four questions guide an honest report.
Use the same chain whenever you choose a measure of centre.
- 1 Purpose + type What should “typical” communicate, and which centres are valid?
- 2 Shape + outliers Is the distribution symmetric, skewed, or extreme?
- 3 Compare centres What does each valid centre represent?
- 4 Report honestly Choose one, justify it, and state the limitation.
What number should represent a “typical employee salary”?
Salary report · Inspect
Purpose, type, and shape come first.
No calculation is useful until the reporting purpose and evidence are clear.
Describe the centre for most employees.
Annual salary, in $000s.
$310K is far above the other seven.
Salary report · Compute + compare
Compute and compare both centres.
Annual salaries in thousands of dollars; the values are already sorted.
\(\sum x_i = 660\)
\(\bar{x}=660/8=82.5\)
$82.5K\(\text{middle values}=50\text{ and }52\)
\(\text{median}=(50+52)/2=51\)
$51KSeven of eight salaries are below the $82.5K mean. The $310K salary creates a long right tail and a $31.5K gap between the centres.
Anonymous written check
In one or two sentences, which centre should lead the salary report, and what evidence makes that choice honest?
Eight annual salaries have mean $82.5K and median $51K. Seven of eight salaries are below the mean because one $310K salary creates a long right tail.
Write one or two sentences. Name the centre and cite the outlier evidence.
Optional · up to 280 characters · do not include names or personal details. Raw text is sent from OttoLearn/Supabase to Anthropic’s Claude to create anonymous class themes; individual responses are never projected. Paper or private writing is equivalent.
Worked salary report · Model
Report the median—and disclose the outlier.
A good conclusion states the choice and the limitation.
The median salary is $51K, so it better represents a typical employee in this right-skewed group.
The mean is $82.5K, but the $310K salary pulls it above seven of the eight employees.
Independent transfer
Your turn: commute-time report.
Commute times in minutes
1. Compute the mean and median.
2. Describe the shape evidence.
3. Choose a primary centre and write two bounded sentences.
Transfer model
Compare with a complete model.
Each reveal answers one step in the decision chain.
Compute: \(\sum x_i=322\), so \(\bar{x}=322/10=32.2\) minutes.
Locate: \(\text{median}=(25+27)/2=26\) minutes.
Inspect: 92 minutes creates a long right tail.
Choose: Report the 26-minute median as typical; disclose the 92-minute commute and the 32.2-minute mean if relevant.
Anonymous check
Which report best describes a typical wait?
Wait times in minutes are 4, 5, 5, 6, 7, 7, 8, 30. The mean is 9 and the median is 6.5.
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Typical wait is 9 minutes because a mean is always the official average
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Typical wait is 6.5 minutes because the 30-minute value creates a right tail
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Typical wait is 7 minutes because 7 is the only valid mode
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No centre exists because one wait is 30 minutes
Anonymous check
Which centre is licensed by these data?
Preferred study location: Library 32, Home 28, Café 14, Other 6.
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Mean = 20 because the four frequencies average to 20
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Median = halfway between Home and Café
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Mode = Library
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No centre because the categories are nominal
Anonymous exit reflection · confidence, not mastery
How confident are you with each idea?
I can compute and interpret a sample mean.
I can find an odd or even median after sorting.
I can explain how an extreme value affects the mean and median.
I can identify unimodal, bimodal, and no-mode cases.
I can choose and defend a measure of centre from type, shape, and outliers.
Return to the opening
The honest rent headline.
Constructed seven-listing example
“Median rent is $920; the mean is $1,650 because one $6,100 penthouse pulls it upward.”
All values are fictional and shown; this is a reporting lesson, not a market estimate.
Main idea · next class
Centre is only part of the story.
Type → shape and outliers → measure → honest claim