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CFA Level I Exam · Statistical Distributions for Financial Asset Prices and Returns

Normal Distribution and Z-scores for CFA Level I

Updated 7 October 2026 · Fact-checked

The normal distribution is a symmetric, bell-shaped distribution fully described by its mean and standard deviation. To solve a question, convert the value to a z-score, z = (x − μ) ÷ σ, then read the cumulative probability from a z-table. Intervals of ±1, ±2 and ±3 standard deviations are the reference ranges.

Understand Normal Distribution and Standardization (Z-scores)

The normal distribution is a continuous, bell-shaped distribution. It is symmetric around its mean, so the mean, median and mode are equal. Two numbers describe it fully: the mean (μ) and the standard deviation (σ). Skewness is 0 and kurtosis is 3 (excess kurtosis is 0).

Because it is continuous, the probability of any single exact value is zero. You only find probabilities for ranges, such as P(X ≤ 5%) or P(−2% < X < 8%). That is why the area under the curve matters.

Standardization turns any normal variable into the standard normal distribution, which has mean 0 and standard deviation 1. The z-score tells you how many standard deviations a value sits above or below the mean. A z of −1.5 means 1.5 standard deviations below the mean. Once you have z, one table serves every normal variable.

The z-table gives the cumulative probability N(z) = P(Z ≤ z). Because the curve is symmetric, N(−z) = 1 − N(z). For probabilities above a value, use 1 − N(z). For a range, subtract the two cumulative probabilities.

A useful rule of thumb gives approximate intervals around the mean: about 68% of observations lie within ±1σ, about 95% within ±2σ (more precisely ±1.96σ), about 99% within ±2.58σ, and about 99.7% within ±3σ. Keep the 99% and 99.7% figures separate: 2.58σ is the 99% interval, and 3σ is a wider interval. Also know the difference between a univariate distribution (one random variable) and a multivariate distribution (several variables together). A multivariate normal distribution is specified by the mean of each variable, each variable's variance, and the correlations (or covariances) between all pairs. Portfolio returns are often modelled this way. A linear combination of normal variables is also normal.

Key formulas to remember

Z-score (standardization)
z = (x − μ) ÷ σ
Counts standard deviations from the mean. Use σ, not σ².
Reverse standardization
x = μ + zσ
Use when a probability is given and you need the cutoff value.
Symmetry of cumulative probabilities
N(−z) = 1 − N(z)
Lets you use positive-z tables for negative z.
Probability above a value
P(X > x) = 1 − N(z)
The table gives the area to the left only.
Probability between two values
P(a < X < b) = N(zb) − N(za)
Standardize both values first.
Approximate confidence intervals
68%: μ ± 1σ; 95%: μ ± 1.96σ; 99%: μ ± 2.58σ
The rule of thumb rounds 1.96σ to ±2σ. Keep 99% (±2.58σ) separate from ±3σ, which covers about 99.7%. Use 1.96 and 2.58 when exact.
Common one-sided z values
N(1.28) ≈ 0.90; N(1.645) ≈ 0.95; N(2.33) ≈ 0.99
Useful for 5% and 1% tail cutoffs.
Linear transformation of a normal variable
If X ~ N(μ, σ²), then aX + b ~ N(aμ + b, a²σ²)
Normal stays normal under linear combinations.

How to solve Normal Distribution and Standardization (Z-scores) questions

Use the same routine for any normal distribution question. It works for probabilities, cutoffs and confidence intervals.

  1. 1Write down μ and σ. Check whether the question gives variance; if so, take the square root.
  2. 2Identify what is wanted: a probability (area), or a value (cutoff) for a given probability.
  3. 3For a probability, convert each boundary to z using z = (x − μ) ÷ σ.
  4. 4Sketch a quick bell curve and shade the required area: left tail, right tail or middle.
  5. 5Find N(z) from the table. Use N(−z) = 1 − N(z) for negative z.
  6. 6Combine: left tail is N(z); right tail is 1 − N(z); middle is N(zb) − N(za).
  7. 7For a cutoff, find the z for the given probability, then compute x = μ + zσ.
  8. 8Check that the answer is sensible: a value above the mean should have probability above 50% to its left.

Quickest way: Shortcut with reference z-values

When to use it: Use when the question asks about 1, 2 or 3 standard deviations, or a common confidence level, so you can avoid the table.

  1. Compute how many σ the value is from μ: (x − μ) ÷ σ.
  2. If the answer is about 1, 2 or 3, use 68%, 95% or 99.7% for the two-sided area. Use 99% only with 2.58σ.
  3. For a one-sided tail, take the two-sided outside area and halve it. Outside 95% is 5%, so each tail is about 2.5%.
  4. Match the result to the three options; often only one fits. Eliminate options that break symmetry or exceed 100%.
  5. On the BA II Plus, there is no direct normal function in the standard Level I workflow, so rely on remembered z-values and the table values given in the exam.

Common mistakes in Normal Distribution and Standardization (Z-scores)

  • Using variance instead of standard deviation in the z-score.

    The question gives σ² and students plug it straight in.

    Fix: Always check the symbol. If you see σ² or 'variance', take the square root first.

  • Reading N(z) as the probability above z.

    Students forget that the table gives the cumulative area to the left.

    Fix: For 'greater than', calculate 1 − N(z). Sketch and shade the area before computing.

  • Mishandling negative z-scores.

    Many tables list only positive z.

    Fix: Use N(−z) = 1 − N(z). For example, if N(1.0) = 0.8413, then N(−1.0) = 0.1587.

  • Treating 68-95-99 as two-sided when the question is one-sided.

    The rule of thumb is quoted as an interval around the mean.

    Fix: For one tail, halve the outside area. About 95% inside means about 2.5% in each tail.

  • Assuming returns are exactly normal.

    The normal is the default model, so students apply it without thought.

    Fix: Remember that real returns often show fat tails and skew. The normal can understate extreme losses.

  • Confusing univariate and multivariate normal.

    Both use the word normal.

    Fix: Univariate needs a mean and variance. Multivariate also needs the correlations between each pair of variables.

Worked examples

Example 1

A fund's annual return is normally distributed with a mean of 8% and a standard deviation of 10%. Given N(1.0) = 0.8413, what is the probability that the return is below −2%? (A) 0.1587 (B) 0.3413 (C) 0.8413

Show the solution
  1. μ = 8%, σ = 10%, x = −2%.
  2. z = (−2 − 8) ÷ 10 = −1.0.
  3. P(X < −2%) = N(−1.0) = 1 − N(1.0).
  4. 1 − 0.8413 = 0.1587.

Answer: (A) 0.1587. The return is 1 standard deviation below the mean, so about 15.87% of outcomes fall below it.

Example 2

Monthly returns on a global equity index are normal with a mean of 1% and a standard deviation of 4%. Given N(1.0) = 0.8413 and N(−0.5) = 0.3085, what is the probability that the return is between −1% and 5%? (A) 0.3085 (B) 0.5328 (C) 0.8413

Show the solution
  1. Lower z = (−1 − 1) ÷ 4 = −0.5.
  2. Upper z = (5 − 1) ÷ 4 = 1.0.
  3. P = N(1.0) − N(−0.5).
  4. 0.8413 − 0.3085 = 0.5328.

Answer: (B) 0.5328. Subtract the left-tail cumulative probability from the upper cumulative probability.

Exam tips

  • Questions are three-option MCQs, so sketch the curve and eliminate options that contradict the area, such as a value above the mean with probability below 50%.
  • Check whether the stem gives variance or standard deviation before computing z.
  • Memorize 1.645, 1.96 and 2.58 as the common cutoffs for 90%, 95% and 99% two-sided intervals, and 1.28, 1.645 and 2.33 for one-sided tails.
  • Expect conceptual items too: symmetry, skewness of 0, kurtosis of 3, and what a multivariate normal needs to be specified.
  • Do not spend more than about 90 seconds; with a z-table supplied in the item, the work is two lines of arithmetic.

Practice questions from Statistical Distributions for Financial Asset Prices and Returns

Normal Distribution and Standardization (Z-scores) in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Normal Distribution and Standardization (Z-scores): frequently asked questions

How do I calculate a z-score and probability from a z-table?

Compute z = (x − μ) ÷ σ. Look up N(z) in the table for the area to the left. For a right tail use 1 − N(z), and for a range subtract the two cumulative values.

What are the 68, 95 and 99 percent confidence intervals for a normal distribution?

About 68% of outcomes lie within one standard deviation of the mean, about 95% within 1.96 standard deviations (roughly 2), and about 99% within 2.58 standard deviations. Within 3 standard deviations the figure is about 99.7%, which is a separate, wider interval. These are two-sided intervals around the mean.

What is the difference between a univariate and a multivariate normal distribution?

A univariate normal describes one random variable with a mean and a variance. A multivariate normal describes several variables together and also needs the correlations between each pair. It is commonly used to model returns on several assets in a portfolio.

Why is the probability of an exact value zero for a normal distribution?

The distribution is continuous, so probability is an area under the curve. A single point has no width, so its area is zero. You always compute probabilities over a range.