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z = (x minus mu) divided by sigma. Positive means above mean. Negative means below.

Z-Scores — Convert to z-score then use the z-table. The table always gives area to the LEFT.

Step by Step

1
The z-score formula
z = (x − μ) / σ — how many standard deviations a value x is from the mean μ. A positive z means x is above the mean; a negative z means below; z=0 means exactly at the mean.
2
Using the z-table for probabilities
The standard z-table gives the area (probability) to the LEFT of a given z-score. To find P(X < x), convert x to a z-score and look up that area directly.
3
Finding the area to the right
Since the total area under the curve is 1, P(X > x) = 1 − P(X < x) — subtract the left-area value from 1 to get the right-area value.
4
Working backward: finding x from a percentile
To find the value of x corresponding to a given percentile, rearrange the z-score formula: x = μ + z×σ, using the z-score that corresponds to that percentile from the table.

Applied Walkthrough

1
A distribution of test scores has Ο = 75 and σ = 8. Find the probability that a randomly selected score is less than 67.
2
Convert to a z-score: z = (67 − 75)/8 = −1.
3
Look up z = −1 in the standard z-table, which gives an area to the left of approximately 0.1587.
4
So P(X < 67) ≈ 0.1587, meaning about 15.87% of scores fall below 67. If instead you wanted P(X > 67), you'd compute 1 − 0.1587 = 0.8413.

Exam Application

Exams test whether you can correctly calculate a z-score, correctly use the z-table (remembering it gives left-tail area), and correctly work backward from a percentile to find the corresponding raw value.

⚠ Common Trap

The most common trap is forgetting that the z-table gives area to the LEFT by default — for a "greater than" probability, you must subtract the table value from 1.

✓ Quick Self-Check

1. What is the z-score formula?
z = (x − μ) / σ.
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2. What does the standard z-table give you directly?
The area (probability) to the LEFT of a given z-score.
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3. How do you find P(X > x) if you know P(X < x)?
P(X > x) = 1 − P(X < x).
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4. For Ο=75, σ=8, what is the z-score for x=67?
z = −1.
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5. How do you find a raw value x from a given percentile's z-score?
x = μ + z×σ.
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