Statisticsdrill

Distributions Updated November 2026

Normal distribution

The normal distribution is the symmetric bell curve described by its mean and standard deviation. To find a probability, convert the value to a z-score, z = (x - mean) / sd, and look up the area under the standard normal curve with a table or NORM.S.DIST.

Syntax

P(X < x) = P(Z < (x - mean) / sd)
PartWhat it is
meanthe center of the curve
sdthe standard deviation, which sets the width
zthe standardized value looked up in the normal table

Examples

IQ scores: mean 100, sd 15. P(X < 115)
z = (115 - 100) / 15 = 1

Result: 0.8413

Excel: =NORM.DIST(115, 100, 15, TRUE)

P(X > 130): z = 2
1 - 0.9772

Result: 0.0228

P(85 < X < 115): z from -1 to 1
0.8413 - 0.1587

Result: 0.6826

Cutoff for the top 5%: z = 1.645
x = 100 + 1.645 * 15

Result: 124.7

Excel: =NORM.INV(0.95, 100, 15)

Common mistakes

  • Reading the table as P(Z > z) when it gives P(Z < z); most tables and NORM.S.DIST give the area to the left.
  • Asking for P(X = 120) exactly; for a continuous distribution that is 0, you need an interval.
  • Passing FALSE to NORM.DIST when you want a probability; FALSE returns the density curve height, TRUE returns the cumulative probability.

Practice Normal distribution

These drills check your answer on live data, free, no account needed.

  • Normal: between two valuesThe fill weight of cereal boxes is normally distributed with mean 482 grams and standard deviation 12 grams. What proportion falls between 478.4 and 495.2 grams? Round to 4 decimal places.
  • Normal: above a valueThe number of kilowatt-hours a household uses per week is approximately normal with mean 60 and standard deviation 6. What is the probability that a randomly chosen household is above 75? Round to 4 decimal places.
  • Normal: percentile cutoffMonthly sales per sales rep are normally distributed with mean 403 units and standard deviation 45. The company rewards the bottom 1% with extra coaching. What sales level marks the cutoff? Round to 2 decimal places.

Questions

What is the standard normal distribution?

A normal distribution with mean 0 and standard deviation 1. Every normal variable becomes standard normal after converting to z-scores, which is why one table serves all cases.

How do I find the value for a given percentile?

Find the z for the percentile (1.645 for the 95th, 1.28 for the 90th) and compute x = mean + z * sd, or use =NORM.INV(percentile, mean, sd) in Excel.

What is 1.96 in the normal distribution?

The z-value that leaves 2.5% in each tail, so 95% of values lie within 1.96 standard deviations of the mean. It is the multiplier in 95% confidence intervals.

Related

  • z-scoreThe z-score formula z = (x - mean) / sd, worked examples, how to read a z-score as a percentile with the normal table, and the Excel function STANDARDIZE.
  • Empirical rule (68-95-99.7)The empirical rule: 68% of normal data lies within 1 standard deviation of the mean, 95% within 2 and 99.7% within 3. Worked examples for intervals and tails.
  • Central limit theoremThe central limit theorem makes sample means approximately normal with sd sigma / sqrt(n). Explanation, worked probability examples and the n = 30 rule.
  • Confidence interval for a meanHow to calculate a confidence interval for a mean with x-bar plus or minus t times s over root n, a full 95% example, when to use z, and Excel CONFIDENCE.T.

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