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.
