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Notes for Normal Distribution:

When computing percentages of data values that fall within a certain range for a normally distributed quantity, we can sometimes use the 68-95-99.7% rule (if the range of values is “right”). The z-score table can be used for ANY problem. To go from a measured value to the z-score, use the following formula:

Standard score = z-score = .

Then using the z-score table, we can read off the percentage. Visually we have the following path:

Measured value compute z-score read off percentage from table

Sometimes we are given a percentile (percentage), and we want to find a cut-off score for the measured values such that the given percentage of data are above or below that value. For this computation, we use the opposite path. First we find the given percentage in the table, then we read off the corresponding z-score. Then we use the formula for the z-score and solve for the measured value.

Find percentage in table read-off z-score compute measured value

Rather than having to solve for the measured value each time, here is the formula for obtaining measured value from z-score:

In both cases, the percentage (or probability) in the table is for values to be BELOW the z-score. You may have to do additional computations to transform the question into that form. This is where shading the area under the curve really helps.

Note: For all problems regarding the Normal distribution, in addition to what is asked, draw the Normal distribution with both the generic scale in terms of standard deviations and the scale that gives the actual measurements. Then shade the area under the curve that corresponds to the percentage that the problem asks for. Also describe how you found your answer and show all your computations, including what you did put into your calculator.

You may use the Normal app (see below) to check on your computations and to see whether you shaded the correct area. Make sure you know how to do the problems using the z-score table, as that is what you need to do in the quiz.

In addition, there are Khan Academy videos that show lots of examples for Normal distribution computations. Click on each link to get a sequence of videos that also let you check for understanding.

· Normal distribution and Empirical rule

· Z-scores

· Normal distribution calculations

To check your answers for Normal computations and to visualize the computations, you can also use a Normal Distribution applet. Here is the URL: https://istats.shinyapps.io/NormalDist/

If you want to find a percentage or probability, select the Find Probability tab.

You can enter the following values:

· the mean and standard deviation for the Normal distribution that you want to use

· the type of percentile/probability to find

· The value of x

The graph will show the area representing the percentile for the given value of x in dark blue. In the example on the right, for a Normal distribution with mean 0 and standard deviation 1, the percentile (or probability) that the values are less than 1.96 is 97.5%.

If you want to find a cut-off value for a given percentage/probability, select the Find Percentile/Quantile tab.

You can enter the following values:

· the mean and standard deviation for the Normal distribution that you want to use

· the type of percentile/probability to find

· The probability

The graph will show the area representing the given percentile in darker blue, and the cut-off value on the horizontal axis in red. In the example below, for a Normal distribution with mean45 and standard deviation 3, the cut-off for the top 10% of the data is 48.845.

Note that only the third entry is different in the two different tabs, and that value reflects the starting point for the path to solve the problem: either the measured value x or the probability.