Z scores and standardization — study notes

What is a z score?

• A z score provides a concise way to describe exactly where an individual score falls within its distribution.

• It represents the number of standard deviations a score is above or below the mean of that distribution.

• A z score is a standardized score: it converts the original unit of measurement (e.g., exam points, temperature) into units of standard deviations.

• Intuition: knowing a score (x) and the mean and spread (SD) of the distribution lets you say how unusual or typical that score is.

• If distributions are roughly normal, z scores help compare scores across different distributions.

Why we care about z scores

• With just a raw score, you often don’t know how extreme it is without the distribution’s mean and spread.

• Two students could have the same raw score (e.g., 78) in different distributions with the same mean but different spreads; z scores reveal relative standing.

• A small standard deviation means scores are tightly clustered around the mean, so a given raw score is more extreme; a large SD means scores are more spread out, so the same raw score is less extreme.

• Z scores summarize three pieces of information in one number: location (relative to mean) and spread (in SD units).

Notation: samples vs populations

• Statisticians distinguish between describing a sample (descriptive statistics) and describing a population (inferential statistics).

• Sample (descriptive): represented with English/ Roman letters (e.g., X, \bar{X}, s) – describe the data you actually collected.

• Population (inferential): represented with Greek letters (e.g., \mu, \sigma, \sigma^2) – describe the full population when known.

• A population parameter (e.g., population mean \mu, population SD \sigma) is often unknown and estimated from sample data.

• Common sample statistics:

  • Mean: (\overline{X}) (often written as M in some contexts)

  • Variance: (s^2)

  • Standard deviation: (s)

• Common population parameters:

  • Mean: (\mu) (mu pronounced like "mew")

  • Variance: (\sigma^2)

  • Standard deviation: (\sigma)

• Pronunciation note: (\mu) is pronounced mu, not like the cow sound "moo".

• One-to-one mapping: mean ↔ (\mu), variance ↔ (\sigma^2), standard deviation ↔ (\sigma).

Formulae: how to compute z scores

• For a score within a sample (descriptive z score):

  • $z = \frac{x - \overline{X}}{s}$

• For a score within a population (inferential z score):

  • $z = \frac{x - \mu}{\sigma}$

• Converting back from a z score to a raw score:

  • Within a population: $x = z\,\sigma + \mu$

  • Within a sample: $x = z\,s + \overline{X}$

• Quick example (population): mean 100, SD 10, raw score 120:

  • $z = \frac{120 - 100}{10} = 2$

• Another example (population): mean 100, SD 10, raw score 85:

  • $z = \frac{85 - 100}{10} = -1.5$

• Another example (sample): mean 4 cups/day, SD 1.5, raw score 6 cups:

  • $z = \frac{6 - 4}{1.5} \approx 1.33$

• Inverse example (to raw from z): z = -2, mean 4, SD 1.5:

  • $x = (-2)(1.5) + 4 = 1$

Worked practice problems from the transcript

• Practice 1 (population parameters):

  • Population mean (\mu = 100), population SD (\sigma = 10), raw score (x = 120):

  • $z = \frac{120 - 100}{10} = 2$

• Practice 2 (population):

  • Mean (\mu = 100), SD (\sigma = 10), raw score (x = 85):

  • $z = \frac{85 - 100}{10} = -1.5$

• Practice 3 (coffee):

  • Mean daily cups = 4, SD = 1.5, raw score = 6:

  • $z = \frac{6 - 4}{1.5} = \frac{2}{1.5} \approx 1.33$

• Practice 4 (inverse):

  • If z = -2, mean = 4, SD = 1.5:

  • $x = (-2)(1.5) + 4 = 1$

• Conceptual practice: interpreting a z score rather than calculating a number

  • A z score of -2.5 (e.g., rainfall) means the value lies 2.5 SDs below the mean, i.e., in the far left tail.

  • In a roughly symmetric normal distribution, moving out to -2.5 SD places you in the far tail, indicating well-below-average value.

Unit conversion intuition: z scores as unit conversions

• Conceptual idea: converting original units to standard deviation units is just unit conversion.

• Examples from the transcript (illustrative):

  • 27 feet ≈ 9 yards (unit conversion)

  • 72°F ≈ 22.22°C (unit conversion; note Fahrenheit to Celsius conversion)

  • 350 liters ≈ 1.43 hogsheads (hogshead is a unit of beer volume)

  • 128 cubic feet of firewood = 1 cord (or also 3 ricks of firewood)

• Takeaway: z scores are simply converting the measurement to the number of standard deviations away from the mean.

Area under the normal curve and z-score interpretation

• Core idea: in a standard normal curve (mean 0, SD 1), you can estimate percentages of scores between or beyond z scores using known segments:

  • From mean to +1 SD: ~34%

  • From +1 SD to +2 SD: ~14%

  • Above +2 SD: ~2%

  • By symmetry, below -1 SD: ~34%; between -1 and -2: ~14%; below -2: ~2%

• Estimating percent below a positive z (e.g., z = +1):

  • Draw the standard normal curve, mark z = +1, shade below it.

  • Area from 0 to +1 is ~34%; area below mean is 50%; total below +1 is ~84%

• Estimating percent above a negative z (e.g., z = -1.5):

  • Shade above z = -1.5; split into regions: above 0 is 50%, 0 to -1 is 34%, left of -1 to -2 is 14%, left of -2 is 2% (estimate the small left tail between -1 and -2 as ~8% in the transcript, noting estimation).

  • Sum: 50% + 34% + ~8% ≈ 92%

• Estimating percent below a negative z (e.g., z = -1.8):

  • Area below -1.8 ≈ 2% (below -2) + a small piece between -2 and -1.8 (estimated ~1–2%) ≈ 3–4%

• Estimating percent above a small positive z (e.g., z = +0.6):

  • Area above +0.6 ≈ small tail above 0.6, combine known segments: between 0 and +1 is 34%, above 2 is 2%, between 0 and +0.6 is part of the 0 to +1 segment; the transcript estimates about 26% above +0.6 using a breakdown of 10% (between 0.6 and 1) + 14% (between 1 and 2) + 2% (above 2).

• Practical note: these are estimates to emphasize understanding; exact values can be obtained with tools.

Exact percentages with calculators and software

• Excel and other tools can compute exact percentages below/above a given z score.

• In Excel (standard normal):

  • Area below z: $\Phi(z) = \text{NORM.DIST}(z, 0, 1, TRUE) \,$

  • Area above z: $1 - \Phi(z) = 1 - \text{NORM.DIST}(z, 0, 1, TRUE) \,$

• You can also use websites or other statistical software for the same results.

• Important concept: If you know the area below z, you can get the area above by subtracting from 1 (since total area under the curve is 100%).

Standardization and cross-distribution comparisons

• Z scores are called standardized scores because they convert raw scores to a common scale (in SD units).

• This standardization allows direct comparisons across distributions with different means and spreads.

• Example: comparing SAT vs ACT performance:

  • SAT: score 680, mean 500, SD 100 → z = \frac{680-500}{100} = 1.8

  • ACT: score 28, mean 18, SD 6 → z = \frac{28-18}{6} = 1.67

  • Since 1.8 > 1.67, the SAT score is further above its mean in SD units, indicating relatively better performance on the SAT than the ACT.

• Another example from the transcript:

  • Drinking: mean 3 drinks/week, SD 2 → last week 6 drinks → z = \frac{6-3}{2} = 1.5

  • Lottery tickets: mean 0, SD 0.5 → last week 2 tickets → z = \frac{2-0}{0.5} = 4

  • The lottery behavior is far more extreme (z = 4) than the drinking (z = 1.5), illustrating cross-distribution comparison.

Summary of key takeaways

• A z score tells you how many standard deviations a value is from the mean:

  • Positive z: above the mean; Negative z: below the mean.

• Z scores are computed differently depending on whether you are describing a sample or a population:

  • Sample: $z = \frac{x - \overline{X}}{s}$

  • Population: $z = \frac{x - \mu}{\sigma}$

• The inverse transformation to get a raw score from a z score is:

  • Population: $x = z\,\sigma + \mu$

  • Sample: $x = z\,s + \overline{X}$

• Z scores enable comparisons across distributions and help interpret how unusual a value is within its distribution.

• You can estimate percentile areas by hand using the standard normal areas; for precise values, use tools like Excel or online calculators.

• Always consider the distribution shape; z-score interpretation assumes approximate normality for the referenced percentiles.

Real-world practice prompts (quick recap)

• If a z score is -2.5, the value is two and a half standard deviations below the mean (far left tail).

• If a z score is +1, about 84% of values fall below that z score (rough estimate: 50% below the mean + 34% from mean to +1).

• If a z score is +0.6, about 26% of the distribution lies above that z score (illustrative estimate using area breakdowns).

• For exact percentages, compute using the standard normal CDF: (\Phi(z)) or software; to get above, use (1-\Phi(z)).

Quick reference formulas

• Z score (sample): $z = \frac{x - \overline{X}}{s}$

• Z score (population): $z = \frac{x - \mu}{\sigma}$

• Raw score from z (sample): $x = z\,s + \overline{X}$

• Raw score from z (population): $x = z\,\sigma + \mu$

• Area below z (standard normal): $\Phi(z) = \text{NORM.DIST}(z, 0, 1, \text{TRUE})$

• Area above z: $1 - \Phi(z)$