Z-Score Detailed Practice

📖 Statistics – Z-Score: Interpretation, Calculation and Comparison

Z-Score — Interpretation, Calculation and Comparison

💡 Why Do We Need a Z-Score?

Imagine: Danny scored 80 in maths and 80 in English.
Are these the same achievement in both subjects?

Not necessarily! If the Maths mean was 70 and English 85, then 80 in Maths is above average, but 80 in English is below average.!

The Z-score solves exactly this problem — it translates every score into a "common language" that allows fair comparison.

What Is a Z-Score?

A Z-score measures how far a value is from the mean, where distance is measured in units of standard deviation — not in points.

🔑 The Core Idea:
A Z-score does not say "how many points you have" — it says "where you stand relative to everyone else"..

Formula

\(z = \dfrac{x - \bar{x}}{S}\)

Where:

\(x\) = individual value (e.g. a student's score)
\(\bar{x}\) = group mean
\(S\) = group standard deviation

⚠️ Note: Z-score is not measured in points— it is measured in "standard deviations from the mean". This is a completely different unit!

How to Interpret a Z-Score?

Z-score	Meaning	Example
\(z > 0\)	Value above the mean	\(z = 1.5\) → 1.5 SDs above mean
\(z = 0\)	Value equal to the mean	Your score is exactly at the mean
\(z < 0\)	Value below the mean	\(z = -2\) → 2 SDs below mean

📌 Rule of thumb: |Z| tells us \(z\) how far the value is from the mean; the sign tells us the direction (above or below).

Example 1 — Basic Z-Score Calculation

📝 Given:

For a certain class:

Mean: \(\bar{x} = 70\)
SD: \(S = 10\)
Dana scored: \(x = 85\)

🔢 Step 1 — Substitute into formula:

\(z = \dfrac{x - \bar{x}}{S} = \dfrac{85 - 70}{10}\)

🔢 Step 2 — Calculate the numerator:

\(85 - 70 = 15\)

🔢 Step 3 — Divide by the standard deviation:

\(z = \dfrac{15}{10} = 1.5\)

✅ Interpretation: Dana is 1.5 SDs above the mean— an excellent result relative to the class!

Example 2 — Negative Z-Score

📝 Given:

In the same class (\(\bar{x} = 70\), \(S = 10\)), Yossi scored: \(x = 55\)

\(z = \dfrac{55 - 70}{10} = \dfrac{-15}{10} = -1.5\)

📌 Interpretation: Yossi is 1.5 SDs below the mean. The \(z\) is negative because the score is below the mean.

Example 3 — Z-Score of Zero

📝 Given:

In the same class (\(\bar{x} = 70\), \(S = 10\)), Michal scored: \(x = 70\)

\(z = \dfrac{70 - 70}{10} = \dfrac{0}{10} = 0\)

📌 Interpretation: Michal is exactly at the mean.. Z-score 0 does not mean the score is zero! It means the score equals the mean.

🎯 Comparing Across Different Groups

Here is the real power of Z-scores! They allow comparison of performance even when means and standard deviations differ..

📝 Full Example — Comparing Subjects:

Danny scored 80 in Maths and80 in English. In which subject is he better relative to the class??

	Maths	English
Danny's score	80	80
Class mean	\(\bar{x} = 70\)	\(\bar{x} = 70\)
SD	\(S = 10\)	\(S = 5\)

Z-score calculation — Maths:

\(z_{\text{מתמ}} = \dfrac{80 - 70}{10} = \dfrac{10}{10} = 1\)

Z-score calculation — English:

\(z_{\text{אנג}} = \dfrac{80 - 70}{5} = \dfrac{10}{5} = 2\)

✅ Conclusion: In English \(z = 2\) vs Maths. \(z = 1\).
Although Danny scored the same raw mark (80), he performs better in English relative to the class because he is further from the mean (2 SDs vs only 1).

💡 Why does this happen?
In English the SD is small (\(S = 5\)), meaning most students cluster near the mean. Scoring 10 points above the mean in a tight group is a bigger achievement than the same gap in a spread-out group..

Common Mistakes

❌ Mistake	✅ Correct
"\(z = 0\) means the score is zero"	\(z = 0\) means the score equals the mean, not that it is zero!
"\(z = -1.5\) means the score is negative"	\(z\) Negative Z means below average.below the mean, not that the raw score is negative
Danny scored 80 in both, so he is at the same level	Need to compare Z-scores, not raw scores
Z-score is measured in points	Z-scores are measured in units of standard deviation

Summary — When to Use Z-Scores?

✅ To know where a value sts relative to the rest
✅ To compare across different groups (subjects, classes, tests)
✅ To identify outliers
✅ To work with the normal distribution and Z-table

Worked Examples

📑 – :
distribution normal ( Z), \(P(Z > 0)\) ?

Show solution

A 0.5000 ✓ Correct

B 0.0000

C 1.0000

D 0.2500

normal (μ=0, σ=1) 0. Yes area find -0 .

:
P(Z > 0) = 0.5.

📑 : P(Z > 1.0)
, \(P(Z > 1.0)\) ?

Show solution

A 0.1587 ✓ Correct

B 0.3413

C 0.8413

D 0.5000

No Z P(Z > z) (area ). z=1.0 0 0.1587.

probability large mean 15.87%.

📑 : P(Z > 1.5)
\(P(Z > 1.5)\) ?

Show solution

A 0.0668 ✓ Correct

B 0.4332

C 0.9332

D 0.5000

: z = 1.5, area 0.0668, -6.7% area.

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