In this explainer, we will learn how to find an unknown mean and/or standard deviation in a normal distribution.

Suppose is a continuous random variable, normally distributed with mean and standard deviation , which we denote by . Recall that we can code by the linear change of variables , where follows the standard normal distribution and for all .

We can also use this process to calculate unknown means and standard deviations in normal distributions. Let us look at an example where we need to find the mean.

### Example 1: Determining the Mean of a Normal Distribution

Suppose is normally distributed with mean and variance 196. Given that , find the value of .

### Answer

In order to find the unknown mean , we code by the change of variables , where the standard deviation . Now follows the standard normal distribution and

We can now use our calculators or look up 0.0668 in a standard normal distribution table, which tells us that it corresponds to the probability that .

Thus,

We can use exactly the same technique to find unknown standard deviations.

### Example 2: Determining the Standard Deviation of a Normal Distribution

Suppose that is a normal random variable whose mean is and standard deviation is . If and , find using the standard normal distribution table.

### Answer

In order to find the unknown standard deviation , we code by the change of variables , where the mean . Now follows the standard normal distribution and

We can now look up 0.0548 in a standard normal distribution table, which tells us that it corresponds to the probability that .

Thus, we have

In the previous examples, we used coding to find an unknown mean or standard deviation when the value of the other parameter was given, along with a probability.

Note that we can find both the mean and the standard deviation simultaneously if two probabilities are given, by solving a pair of simultaneous equations. Here is an example of this type.

Example 3: Determining the Mean and Standard Deviation of a Normal Distribution

Let be a random variable that is normally distributed with mean and standard deviation . Given that and , calculate the values of and .

### Answer

In order to find the unknown mean and standard deviation , we code by the change of variables . Now follows the standard normal distribution and and

Using our calculators or looking up 0.6443 and 0.9941 in a standard normal distribution table, we find that these are the probabilities that and that .

This yields the pair of simultaneous equations and We multiply both equations by :

Then, we subtract the second from the first to get

Therefore, we have

We can now substitute back into the equation , which gives us

We arrive at values of and , to the nearest integer.

We can use this method of simultaneous equations to find other unknown quantities in normal distributions.

### Example 4: Finding Unknown Quantities in Normal Distributions

Consider the random variable . Given that and , find the value of and the value of . Give your answers to one decimal place.

### Answer

In order to find the unknown standard deviation and constant , we code by the change of variables , where the mean is . Now follows the standard normal distribution and and

Using our calculators or looking up 0.1 and 0.3 in a standard normal distribution table, we find that these are the probabilities that and that . This yields the pair of simultaneous equations and

We multiply both equations by :

Then, we multiply the second of these by 2:

We can now eliminate by subtracting the second equation from the first:

To find the value of , we can substitute back into :

Thus, rounding to one decimal place, we have and .

Let us try applying these techniques in a real-life context to find an unknown mean.

Example 5: Determining the Mean of a Normal Distribution in a Real-Life Context

The heights of a sample of flowers are normally distributed with mean and standard deviation 12 cm. Given that of the flowers are shorter than 47 cm, determine .

### Answer

We have a normal random variable with unknown mean. To convert the population percentage of into a probability, we divide by 100, so we have .

In order to find the unknown mean , we code by the change of variables , where the standard deviation is . Now follows the standard normal distribution and

We can now use our calculators or look up 0.1056 in a standard normal distribution table, which tells us that it corresponds to the probability that . Thus, we have giving us to the nearest integer.

We can also find unknown standard deviations in real-life contexts.

Example 6: Determining the Standard Deviation of a Normal Distribution in a Real-Life Context

The lengths of a certain type of plant are normally distributed with a mean and standard deviation . Given that the lengths of of the plants are less than 75 cm, find the variance.

### Answer

We have a normal random variable with unknown variance. To convert the population percentage of into a probability, we divide by 100, so we have .

In order to find the unknown variance , we code by the change of variables , where the mean . Now follows the standard normal distribution and

We can now use our calculators or look up 0.8413 in a standard normal distribution table to find that this is the probability that . Thus, we have

Therefore, our variance is , to the nearest integer.

Let us finish by recapping a few important concepts from this explainer.

### Key Points

- Given a normal random variable and a probability , we can code by the change of variables , where . Then, we can use the standard normal distribution to find an unknown mean or standard deviation.
- If we are given two probabilities and , then we can derive a pair of simultaneous equations to find the mean and the standard deviation when both are unknown.
- We can use these techniques to solve real-world problems involving unknown means and standard deviations in normal distributions.