Normal distributions

This is about probability of measuring some continuous quantity, such as time, length, and mass, to be within some interval.

Contents

• When to Use
• Definition
• Mean = Median = Mode
• Empirical rule and $z$-score
• Examples
• Tips

When to Use

If it exhibits the following symptoms, normal distribution may be for you

• a single measurement or occurrence
• no reference to number of tries or measurements
• measurement is continuous (time, length, mass etc)

Normal distribution questions often state the use of the normal distribution. However, it is often the case that a question starts out normal, only to shift into binomial distribution regarding $k$ successes out of $n$ tries.

Definition

A normally distributed random variable $X$ is notated as

with expected value (or mean) $\mu$ and standard deviation $\sigma > 0$ or variance $\sigma^2$.

Normal distribution involves a continuous distribution, meaning the probability can only be calculated over an interval. The probability of any particular value is $0$.

This is done on the calculator. For probability where the interval involves $-\infty$ or $\infty$, use $-1000$ or $1000$, or any arbitrary values more than $4$ standard deviations away from the mean in the correct direction.

To find the endpoint $x$ such that $\text{P}(X < x) = 0.4$, take the inverse normal using $p=0.4$ and the given mean and standard deviation. Calculators often use standard deviation, while the mathematical definition uses variance.

For values within distance $d$ of $k$, you may see

For values outside of distance $d$ of $k$, you may see

Mean = Median = Mode

For a normal distribution, $\mu$ is the mean, the median, and the mode. This means

This also means that normal distribution is symmetric about the mean.

Other properties based on the symmetry include, for some $0 < j < k$

Empirical rule and $z$-score

The empirical rule, or $68-95-99.7$ rule, states that, for a normal distribution, regardless of the values of $\mu$ or $\sigma$

For a single measurement, there is a $68\%$ chance it is within $1$ standard deviation of the mean, $95\%$ chance of within $2\sigma$, and $99.7\%$ chance of within $3\sigma$.

The above are generally not true for other types of distributions.

This prompts the definition of the $z$-score

The $z$-score is the number of standard deviations that a value $x$ is more than the mean. Negative $z$-scores mean the value is that many standard deviations below the mean.

Then all calculations can simply refer to the standardized normal distribution $\text{N}(0, 1)$.

$z$-scores allow for solving for $\mu$ or $\sigma$ when given the probability.

A sketch of the derivation is provided in the discussion on definite integrals.

Examples

On TI-84 Plus and TI-Nspire, the inverse normal function invNorm returns $x$, such that $\text{P}(X < x) = p$ for some known distribution and probability $p$. On Casio and other brands, there is often an option to solve $\text{P}(X > x) = p$ as well. We shall assume we can only solve the first case on the calculator.

Tips

• Draw a bell-curve. Label the mean and the desired interval(s).
• Which way is the inequality sign?
• Does it want probability of being inside or outside some interval?
• Does it want a minimum or a maximum?
• Math notation uses variance but calculators typically use standard deviation.
• Write down the calculator command you used, so you can easily check your work later.
• The intersection of two events is their area(s) of overlap. Conditional probability is a ratio (fraction) of areas. See also probability formulas.