Standard Normal Distribution and z-score
To calculate probabilities for values that are not exactly \(1\) or \(2\) standard deviations from the mean, we must convert the graph of our normal distribution to the standard normal distribution which has a mean of \(0\) and standard deviation of \(1\). To do this, we must calculate the z-score which corresponds to this distribution. The sign (positive or negative) of the z-score will tell us if we are above or below the mean.
A z-score represents the distance a data value is from the mean, so if \(z = 1\), the data value is one standard deviation above (to the right of) the mean. If \(z = -2.3\), the data value is \(2.3\) standard deviations below (or to the left of) the mean.
To calculate probabilities for values that are not exactly \(1\) or \(2\) standard deviations from the mean, we must convert the graph of our normal distribution to the standard normal distribution which has a mean of \(0\) and standard deviation of \(1\). To do this, we must calculate the z-score which corresponds to this distribution. The sign (positive or negative) of the z-score will tell us if we are above or below the mean.
A z-score represents the distance a data value is from the mean, so if \(z = 1\), the data value is one standard deviation above (to the right of) the mean. If \(z = -2.3\), the data value is \(2.3\) standard deviations below (or to the left of) the mean.
Converting a data value to a z-score: \(z=\large\frac{x-\mu}{\sigma}\)
where \(x\) represents the data value, \(\mu\) represents the mean and \(\sigma\) represents the standard deviation.
where \(x\) represents the data value, \(\mu\) represents the mean and \(\sigma\) represents the standard deviation.
Each z-score has a specific area under the curve associated with it, which can be located in a z-table. There are several formats for z-tables, so it is important to understand how the table you are using is designed.
Using the Standard Normal Table:
Once you have calculated the z-score, look up that value in the normal table.
z-tables
Steps for Calculating Probabilities using Z-scores:
1. Write a probability statement in terms of X.
2. Ask yourself…is the z-score provided? If yes, continue to step #3. If no, convert X into a z-score.
3. Rewrite the probability statement in terms of z.
4. Draw the standard normal curve, plot the value(s) of z, shade the desired region.
5. Use the table to determine the shaded area.
6. Answer the question in context.
Example 1: Data collected over a period of years shows that the average daily temperature in Honolulu is \(73^{\circ}\) F with standard deviation of \(50^{\circ}\) F. The data is approximately normally distributed. Find the following probabilities:
The temperature is between \(65^{\circ}\) F and \(85^{\circ}\) F.
The temperature is greater than \(61^{\circ}\) F.
The temperature is less than \(87^{\circ}\) F.
Using the Standard Normal Table:
Once you have calculated the z-score, look up that value in the normal table.
- Center Area Table: The entries in the table correspond to the area between the mean, 0, and the z-score.
- Left Tail Table: The entries in the table correspond to the area to the left of the z-score.
z-tables
Steps for Calculating Probabilities using Z-scores:
1. Write a probability statement in terms of X.
2. Ask yourself…is the z-score provided? If yes, continue to step #3. If no, convert X into a z-score.
3. Rewrite the probability statement in terms of z.
4. Draw the standard normal curve, plot the value(s) of z, shade the desired region.
5. Use the table to determine the shaded area.
6. Answer the question in context.
Example 1: Data collected over a period of years shows that the average daily temperature in Honolulu is \(73^{\circ}\) F with standard deviation of \(50^{\circ}\) F. The data is approximately normally distributed. Find the following probabilities:
The temperature is between \(65^{\circ}\) F and \(85^{\circ}\) F.
The temperature is greater than \(61^{\circ}\) F.
The temperature is less than \(87^{\circ}\) F.
You can also find the probability using the graphing calculator. Steps 1 - 4 will remain the same, but we can use the calculator instead of the z-table to find the probability. Take a look at the flowchart below, then watch the video to see Example 1 completed with the calculator.
Lower and Upper represent the left and right z-scores. For positive infinity use \(10^{99}\) and for negative infinity use \(-10^{99}\).
Z-scores can be useful when comparing two sets of data with different means and standard deviations. For example, comparing performance on the ACT vs SAT.
Example 2: The national average ACT composite score is \(20.8\) with a standard deviation of \(4.8\). The national average SAT combined score is \(1500\) with a standard deviation of \(250\). Both scores are approximately normally distributed. Sue scored a \(29\) on her ACT. Paul scored an \(1820\) on his SAT. Who performed better?
Sue performed better than Paul because her score placed her \(1.71\) standard deviations above the mean compared to Paul’s score, which placed him \(1.28\) standard deviations above the mean.
It is important to note that if your z-score is negative, you want to be closer to the mean not farther from it. So be careful when doing comparisons with z-scores!
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