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A quick, calm intro — then you choose when to climb.
Statistics reads a list of numbers back to you. Four summaries, each answering a different question.
Mean — spreads the total evenly — add every value, divide by how many there are (the average)
Median — the middle value — but only after sorting from smallest to largest
Mode — the value that appears most often
Range — biggest − smallest — how far the data stretches
Every step explained — why we do it, and what happens.
Why: These five numbers are the whole dataset — every summary below is built from them. Write the list down first, then calculate.
What happens: The data: {12, 8, 15, 12, 13}.
Why: The mean spreads the total evenly: add every value, divide by how many there are.
What happens: Mean of {12, 8, 15, 12, 13} is 12.
Why: The median and mode need order and counting — sort from smallest to largest first.
What happens: Sorted: 8, 12, 12, 13, 15.
Why: With five values, the median is the 3rd — the middle of the sorted list.
What happens: Median of {12, 8, 15, 12, 13} is 12.
Why: The mode is the value that appears most often — 12 shows up twice, everyone else once.
What happens: Mode is 12.
Why: The range measures the spread: biggest − smallest. Only the extremes matter.
What happens: Range is 7.
Box Plots & IQR
The 5-number summary that visualizes data spread — from the same five scores as the lesson.
How the numbers are found
Sort first — the same sorted list as the lesson
Five numbers: the middle one (12) is the median. It splits the list into a lower half and an upper half.
Q1 — the median of the lower half
Lower half: 8, 12. With two numbers, the median is their average.
Q3 — the median of the upper half
Upper half: 13, 15. Same move — average the two.
IQR — how far apart the quartiles are
The middle 50% of the data sits between Q1 and Q3.
The plot — every number comes straight from the steps above: min 8 · Q1 10 · median 12 · Q3 14 · max 15.
The basketball team scored 12, 8, 15, 12 and 13 points in five games. The coach wants the “typical” score: find the mean, the median and the range.
Why: The mean starts with the total — add all five scores.
What happens: Total = 60 points.
Why: Divide the total by the number of games: 60 ÷ 5.
What happens: Mean = 12 points per game.
Why: The glitch: grabbing the middle of the UNSORTED list. Sort first: 8, 12, 12, 13, 15 — now the middle is 12.
What happens: 15 was just the fourth game in the list, not the middle.
Why: The sorted middle is 12; the range is biggest minus smallest: 15 − 8.
What happens: Median 12, range 7.
Why: The check: a mean of 12 across five games must total 5 × 12 = 60 — exactly the sum we started with.
What happens: 60 = 60 ✓ — the mean is right.
Two classic glitches:
1.Median without sorting
Wrong:The median is the middle number of the list as written
Right:Sort from smallest to largest first — the median is the middle of the SORTED list.
2.Mean vs median mix-up
Wrong:Mean and median are two names for the same average
Right:The mean spreads the total evenly (add ÷ count); the median is the sorted middle. One huge outlier drags the mean, not the median.
You've got the concept! Let's lock it in with practice.
Learning is optional — the ladder is always open.