You can calculate the expected hypothetical distribution with a hand calculator.
You may wish to see whether you have obtained an unexpectedly high or low result, say, in a marathon race situation. To determine that, you can compare the results you actually received with the results you would have expected to get. Those expected results can be found by calculating the expected hypothetical distribution of outcomes, for example, the standard normal curve.
Instructions
Preparation
1. Count the number of cases for the data. If you have race figures for 2,000 runners, you have 2,000 cases.
2. Calculate the mean and standard deviation for the data. The mean is the arithmetic average. The standard deviation is a statistic of the dispersion of the data and can be calculated either with a hand calculator, a statistical software program or a spreadsheet program.
3. Make a two-axis graph on the graph paper, with the x-axis running from the smallest to the highest value for your data. If the fastest runner came in at 1 hour, the lowest value of the x-axis should be about 30 minutes. If the longest runner came in at 8 hours, the highest value of the x-axis should be about 7.5 hours.
4. Make the y-axis frequency run from 0 to about 40 percent of the number of cases. If you have 2,000 runners, the y-axis should run from 0 to about 40 percent of 2,000, or 800.
5. Draw a vertical line on the x-axis at the mean value, which you calculated in Step 2.
Draw the Hypothetical Distribution
6. Place the diagram of the normal curve near your graph to use as a guide.
7. Place the high point of the normal curve on the vertical line at the mean value, up near the top of the y-axis.
8. Begin to draw the downward curve on the right side of the normal distribution. Place the location corresponding to "+1 standard deviation" over the x-axis at the numerical point equal to the mean plus the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "+1 standard deviation point" of the normal curve is above the numerical point 5 on the x-axis, about 60% up the way to the level of the highest point.
9. Continue the downward curve on the right side of the normal distribution. Place the location corresponding to "+2 standard deviations" over the x-axis at the numerical point equal to the mean plus twice the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "+2 standard deviations" point of the normal curve is above the numerical point 6 on the x-axis, about 20% up the way to the level of the highest point.
10. Conclude the downward curve on the right side of the normal distribution. Place the location corresponding to "+3 standard deviations" over the x-axis at the numerical point equal to the mean plus three times the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "+3 standard deviations" point of the normal curve is above the numerical point 7 on the x-axis, about 10% up the way to the level of the highest point.
11. Begin to draw the downward curve on the left side of the normal distribution, which will be a mirror reflection of the right side. You will place the location corresponding to "-1 standard deviation" over the x-axis at the numerical point equal to the mean minus the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "-1 standard deviation point" of the normal curve is above the numerical point 3 on the x-axis, about 60% up the way to the level of the highest point.
12. Continue the downward curve on the left side of the normal distribution. Place the location corresponding to "-2 standard deviations" over the x-axis at the numerical point equal to the mean minus twice the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "-2 standard deviations" point of the normal curve is above the numerical point 2 on the x-axis, about 20% up the way to the level of the highest point.
13. Conclude the downward curve on the left side of the normal distribution. Place the location corresponding to "-3 standard deviations" over the x-axis at the numerical point equal to the mean minus three times the standard deviation. If your mean time is 4 hours and your standard deviation is 1 hour, the "-3 standard deviations" point of the normal curve is above the numerical point 1 on the x-axis, about 10% up the way to the level of the highest point.
Calculate the Portions of the Expected Hypothetical Distribution
14. Draw vertical lines on the x-axis at the positions for the mean plus and minus 1, 2, and 3 standard deviations. In the example in the preceding Section, you would draw vertical lines at the numerical points 1, 2, 3, 5, 6, and 7 on the x-axis.
15. Write in the hypothetical percentages under the normal curve for each segment of the curve. These are as follows for the right side of the curve: between the mean and plus 1 standard deviation, 34.13%; between plus 1 standard deviation and plus 2 standard deviations, 13.59%; between plus 2 standard deviations and plus 3 standard deviations, 2.15%; above plus 3 standard deviations, 0.13%. Write in the corresponding equivalent percentages on the left side of the curve.
16. Calculate and write in the expected absolute numbers expected within each segment of the curve. This is equal to the total number of cases times the hypothetical percentages for each segment. For example, on the right side of the curve, if you have 2,000 cases, between the mean and plus 1 standard deviation you would expect 2,000 times 34.13%, or 682.6 cases.
Tags: standard deviation, standard deviations, numerical point, normal curve, plus standard, point normal