Finding an x-value from a probability

The previous pages explained how to find the probability that a value from a normal distribution will be less than some value, x.

In some circumstances, we must solve the inverse problem — we are given a probability and must find the value x such that there is this probability of being less.

Terminology

Quartiles
The quartiles of a distribution are the three values such that there is probability 1/4, 2/4 and 3/4 of being lower.
Percentiles
The r'th percentile of the distribution is the value with probability r/100 of being lower.
Quantiles
These are generalised by the term quantile. The value with probability p of being lower is called the quantile of the distribution corresponding to probability p.

Weights of apples

The diagram below shows the distribution of weights of Fuji apples arriving at a packhouse. The distribution is normal (µ = 180g, σ = 10g).

The slider translates apple weights, x, into z-scores and uses the z-scores to find the probability of getting an apple with weight less than x.

The largest 10% of apples will be sold for export. How large will these apples be?

This question wants the weight, x, such that

P ( Apple weight < x )   =   0.9

Adjust the slider to make the probability 0.9.

Finding quantiles

The above 'trial-and-error' method of finding a quantile involves trying different x-values until the target probability is attained.

x z-score probability

A better method performs the inverse operations directly,

probability z-score x

The first step of this process involves finding the z-score for which there is the required probability of being less. Statistical software or Excel can evaluate this z-value, or statistical tables can be used. For example, the diagram below shows how to find the z-score such that there is probability 0.9 of being less.

Translating from a z-score to the corresponding x-value is done with the formula,

x  =  μ  +  z σ

(Remember that the z-score tells you how many standard deviations you are from the mean.)

Weights of apples

The diagram below shows the distribution of weights of Fuji apples arriving at a packhouse. The distribution is normal (µ = 180g, σ = 10g).

Use the slider to find the z-score corresponding to any probability. (The computer does the calculation, but normal tables could alternatively be used as described above.)

The z-score is then translated into an apple weight.