Mean, Standard Deviation, and Resistance

Lesson 1 of 2: Compare Center and Spread

In this lesson:

• Build the standard deviation step by step
• Discover why median and IQR resist outliers

Learning Objectives for This Unit

By the end of this unit, you should be able to:

1. Compute the mean, median, IQR, and SD
2. Judge shape and select fitting statistics
3. Explain why median and IQR resist outliers
4. Compare data sets with matched statistics
5. Write comparisons of center and spread

Two Classes Share One Average

Both classes scored a mean of 8 on the same quiz.

• Class A: scores bunched tightly near 8
• Class B: scores scattered from low to high

Same center — but not the same class. We need a number for the spread.

Mean as the Balance Point

For the set 4, 6, 8, 10, 12:

The mean is the balance point — where the data would balance on a seesaw.

Now ask: how far is a typical value from 8?

Standard Deviation Means Typical Distance

The standard deviation (SD) is the typical distance a value sits from the mean.

• Small SD → values huddle near the mean
• Large SD → values spread far from the mean

It is a distance: always positive, same units as the data.

Standard Deviation: Steps One and Two

Step 1 — deviations from 8:
Step 2 — square them:

Standard Deviation: Steps Three and Four

Step 3 — average the squares (the variance):

Step 4 — take the square root:

Say the Standard Deviation in Words

For 4, 6, 8, 10, 12, the SD is about 2.83:

The values sit about 2.83 units from the mean of 8, on average.

• For large data sets, a calculator or spreadsheet does the arithmetic
• Your job is knowing what the number means

Same Mean But Different Spread

• Both sets have mean 8
• Top SD ; bottom SD — bottom is more consistent

Quick Check: Which Is More Spread?

Two sets, both with mean 8:

• Set 1: SD
• Set 2: SD

Which is more spread out?

Think before advancing.

Answer: Set 1 — a larger SD means values sit farther from the mean. Bigger SD means more spread, not bigger or better.

Your Turn: Build the SD

For the set 2, 4, 6, 8, 10:

1. Find the mean
2. Find each deviation, then square it
3. Average the squares
4. Take the square root — say what it means

Work all four steps, then advance.

Answer: Mean ; squares ; variance ; SD .

The Standard Deviation Used Every Value

Building the SD relied on the size of every value:

• Deviations, squares, the average — all depend on magnitude

So here's the question that drives the rest of this lesson:

What happens to these statistics if one value goes haywire?

Recall the Median and the IQR

From the last unit, for ordered data:

• Median — the middle value (for 4, 6, 8, 10, 12 it is 8)
• Five-number summary — min, Q1, median, Q3, max
• IQR — width of the middle 50%

These depend on position, not size.

Predict: Change the 12 to a 60

Keep 4, 6, 8, 10, and change 12 → 60.

Predict how each statistic responds:

• Mean — big change or small?
• Median — big or small?
• SD — big or small?
• IQR — big or small?

Commit to a prediction for each, then advance.

Reveal: Mean Jumps, Median Holds

• Mean: — dragged far right
• Median: — unmoved

Is 17.6 a good "typical" value? No — most values are 10 or below.

Reveal: SD Balloons, IQR Steady

After changing 12 → 60:

• SD: over 20 — balloons
• IQR: nearly unchanged — the middle 50% never saw the 60

One spread statistic panics at the outlier; the other stays calm.

Which Statistics Resist the Outlier?

Matched pairs: mean with SD, median with IQR.

Why the Median Held Steady

The median reads position; the mean reads magnitude.

• To the median, 60 is just "the largest value" — same role 12 had
• Its size is irrelevant; it could be 60 or 6000

Position-based statistics resist outliers because size doesn't change position.

The Range Is Maximally Non-Resistant

The range = max − min uses only the two most extreme values.

• For 4, 6, 8, 10, 12: range
• Change 12 → 60: range , but IQR barely moves

The range lives among the very values an outlier hides in.

Your Turn: The Outlier Stress Test

For 10, 12, 14, 16, 18, then change 18 → 90:

1. Note the mean, median, SD, and IQR
2. Recompute after the change
3. Report which moved and which held

Predict using the pattern, then check.

Answer: Mean and SD jump; median and IQR hold again.

Full Task: Compute, Then Disrupt

For 5, 7, 9, 11, 13:

1. Compute the mean, median, SD, and IQR
2. Change 13 → 50 and recompute all four
3. Which pair stayed resistant?

Do the whole task unaided, then advance.

Answer: Mean 9 → 16.4 and SD balloon; median 9, IQR 4 hold.

Key Takeaways From This Lesson

✓ SD is the typical distance from the mean

✓ Median and IQR resist outliers; mean and SD do not

✓ Use matched pairs: mean–SD, median–IQR

SD measures spread only — not size

The range uses only two extremes

Next: which pair to report, and comparing fairly.