Chapter 5: Displaying and Describing Quantitative Data

Standard Deviation

It’s like a ruler to measure how spread out data is. The bigger the standard deviation, the more scattered the values are.

We use it to compare how far a value is from the average.

Z- Score

It tells you how far a value is from the mean — in standard deviations.

Formula:

z = \frac{x - \mu}{\sigma}

• If z = 0 → the value is the mean

• If z = 2 → it’s 2 SDs above the mean

• If z = –1.5 → it’s 1.5 SDs below the mean

Why do we standardize data?

So we can:

• Compare values from different groups or units (e.g., height vs. weight)

• Identify unusual values (big or small z-scores)

• Work with the Normal Model (aka bell curve)

What happens when we shift data (add/subtract)?

Example: add 10 to every value.

• The center (mean, median) changes

• The spread (IQR, SD) stays the same

What happens when we rescale data (multiply/divide)?

Example: convert kg to lbs (multiply by 2.2).

• Everything changes — mean, median, IQR, range, and SD all get multiplied

• The shape stays the same

What does standardizing do to a distribution?

• Makes mean = 0

• Makes standard deviation = 1

• Shape doesn’t change (still skewed if it was skewed)

How big is a “big” z-score?

No official rule, but:

• |z| > 2 is usually considered unusual

• |z| > 3 is very rare
  Always look at context though — what’s “big” in one dataset might be normal in another

When can I use the Normal Model?

Only when the data is:

• Unimodal (one peak)

• Symmetric

• No extreme outliers
  Check with a histogram or Normal probability plot

What is the Normal Model (bell curve)?

It’s a model that applies to symmetric, unimodal data.

We write it as:

N(μ, σ) → mean and standard deviation

After converting data to z-scores, we use the Standard Normal Model:

N(0, 1)

68–95–99.7 Rule

This helps estimate probabilities:

• About 68% of data is within 1 SD of the mean

• 95% is within 2 SDs

• 99.7% is within 3 SDs

How do I find probabilities or areas under the curve?

You have 3 options:

• Z-table (Table A on the AP exam)

• Calculator:
  

  • normalcdf(lower, upper, μ, σ)

  • invNorm(area, μ, σ) to go backward (from % to score)

• 68–95–99.7 Rule for quick estimates

How do I go backwards from a percentile?

Use invNorm or Table A.

Example: Find the IQ that’s in the top 10%

• Look for area = 0.9000 → z ≈ 1.28

• Use:
  x = \mu+ z\sigma