Using the normal distribution

The normal distribution models a continuous quantity, something measured rather than counted, such as height, mass, time, or a test score, whose values cluster symmetrically around a mean, thinning out towards the extremes to give the familiar bell-shaped curve. We write $X\sim N(\mu,\sigma^{2})$, where $\mu$ is the mean and $\sigma$ is the standard deviation.

Because there is a different bell curve for every $\mu$ and $\sigma$, we do not have a table for each one. Instead we convert any normal variable into the single standard normal variable $Z$, which has mean $0$ and standard deviation $1$, using the Z-score $Z=\frac{X-\mu}{\sigma}$.

One standard table then answers every question, letting us find the probability that $X$ lies above a value, below a value, or between two values.

Reach for the normal distribution when the variable is a continuous measurement and the question states, or clearly implies, that it is normally distributed with a given mean and standard deviation. Signal words include 'normally distributed', 'mean', 'standard deviation', and a request for the probability that a measurement exceeds, falls below, or lies between certain values.

This is the opposite situation to the binomial distribution, which counts successes in a fixed number of trials. If you are counting whole 'successes', think binomial; if you are measuring a quantity that can take any value on a scale, think normal.

The tell-tale first move for a normal question is always to standardise, convert the given $X$-value into a $Z$-score before touching the table.

1. 1

   Write down the parameters

   Note the mean $\mu$, the standard deviation $\sigma$, and the $X$-value(s) in the question.

2. 2

   Standardise

   Convert each $X$-value to a Z-score with $Z=\frac{X-\mu}{\sigma}$.

3. 3

   Sketch and shade

   Draw the standard normal curve and shade the region whose probability you want.

4. 4

   Read the table

   Look up the table value for the relevant $z$. The SPM table gives the upper-tail area $P(Z>z)$.

5. 5

   Use symmetry or complement

   Adjust with the total area $1$ and the curve's symmetry to get $P(Z<z)$ or a 'between' probability.

6. 6

   State the probability

   Write the final probability, matching it back to the original $X$ description.

• Forgetting to standardise and trying to read the table with the raw $X$-value.
• Dividing by the variance $\sigma^{2}$ instead of the standard deviation $\sigma$ when finding $Z$.
• Sign slips when $X<\mu$: the Z-score should be negative, and its size still matters.
• Confusing 'area to the left' with 'area to the right', know exactly which tail your table gives.
• For a 'between' probability, subtracting the wrong two areas instead of sketching first to see what to combine.

The step that costs the most marks is the shading: students standardise correctly, then take the wrong area from the table because they never drew the curve. In a one-to-one lesson our teachers insist on a quick sketch every time and talk you through which region matches $P(X>k)$, $P(X<k)$ or a 'between' probability, so symmetry and complements stop feeling like guesswork.

We also check that you divide by $\sigma$, not $\sigma^{2}$. Teachers at spmaddmath.com.my are experienced, and lessons are online and taught in English.

To see how we teach the normal distribution, message us on WhatsApp to arrange a one-hour paid class from RM50/hr at the teacher's rate.