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## Statistics and probability

### Course: Statistics and probability>Unit 4

Lesson 5: Normal distributions and the empirical rule

# More empirical rule and z-score practice (from ck12.org)

More Empirical Rule and Z-score practice. Created by Sal Khan.

## Want to join the conversation?

• It seems to be completely arbitrary when one must subtract the z-score table value from 1. Sometimes you do both numbers, sometimes neither, sometimes only one of the two. What am I missing?? • When you're starting (and sometimes after you're used to it), it helps a lot to draw a picture of a bell curve and shade in the part that you're trying to measure. Remember (and they usually draw a picture on the z-score table to drive this point home) that the z-score measures the whole area from your point to the left. So that's great if you want the left tail of a distribution, but if you want a right tail then you need to calculate the left tail and subtract it from 1 (which is the area of the entire standard normal curve). And if you want the area between two points, you need to calculate the left tail of the higher number and then subtract the left tail of the lower number to rule out the part of the curve that you don't want to measure. And if you want to measure the percentage of both tails, then you need to measure the left tail of the higher number, subtract that from one to give you the right tail of the higher number, and then add in the left tail of the lower number to get the area you want from the z-score table.
• Sal and Team, why is a z-table value (that corresponds to a z-score) represented as say 0.5786 instead of the final 57.86% we have to give as an answer in the exercises? Why not represent the percentage value from the get go, so we don't have to multiply it by a hundred to get the % value? Just curious. Thanks. • I think it's mostly because the z-score is an intermediate step in a math problem, and the custom of expressing decimals as percentages is an end-of-the-problem step to put the probability in a form that is familiar to people. If tables gave z-scores as percentages, for most of the problems you did you would have to turn them back into decimals to finish whatever problem you were solving.
• Why does the actual z score computed from the table differ from the percentage calculated using the empirical rule? P(Z > 2) is = .5 - P(0 <= Z <= 2) = .5 -.4772 = .0228 which is 2.28% not 2.5%. Did I make a mistake in calculations? I understand if this is a little pedantic but my mathematics professors demand precision and 2.3% is not equal to 2.5%. Any clarification would be greatly appreciated. • How can we use info like z-scores and averages to calculate the probability of success or failure?
For instance I know the average proficiency and average rate of improvement in mathematics of a batch of students, and I want to know what is the probability that they will pass a math exam in the near future or the percentage of students that will most likely pass that exam. • I have a problem learning something and i dont know what video i need to watch.
An example question from my stat. book says "find the indicated area under the standard normal curve - To the left of z = 1.54" How do i do this? Thanks! -Avery • What you need is a z-table. That lists the area of the bell curve to the left of a z-value. Many z-tables only lists for positive z-values, but if you have a negative one you can work out the area anyway. Here's a good one though that shows negative z-values too: http://lilt.ilstu.edu/dasacke/eco148/ztable.htm

In your case you would scroll down until you see the 1.5 in the first column, and then go 5 steps to the right, to the 0.04 column to add the 1.50 + 0.04 = 1.54. In that box you see the area of the curve to the left of z=1.54, which is exactly what you were looking for =) to help you find the right one, it says 0.9382.
• What does Z scores 3 mean?
(1 vote) • what if the numbers that are given to me arent as perfect as in these examples? say if they ask for a kid as tall as 160 cm, how would i measure that? • In the CFA level 1 text books it says you cannot use the Z-Score if the distribution is Nonnormal, variance is unknown (in the case of sal's example variance was known) and if the sample size is smaller than 30 (n<30). I guess if Sal was using population instead of a sample the answer to the first question would be correct, has anyone come across this before? • I cannot figure out how to answer this question from any of these videos:

Find the Z-scores that separate the middle 68% of the distribution from the area in the tails of the standard normal distribution. Can someone please explain? • I'm learning this right now, and I'm learning about probability with normal distribution. Is there a formula where you give it the z-score and it returns the probability? I have to look on a chart, but its hard to use.
(1 vote) • There is no direct formula, no. The table was constructed precisely for that reason. It is impossible to express the probability as a simple formula that you can plug a number into.

That being said, there are alternatives to the table. Many scientific calculators have a normalcdf command (I use a TI-84). Common spreadsheets such as MS Excel, Google Spreadsheets, etc, also have commands which will calculate the probability for you.

For any of these ways to bypass the table, however, it is still important to understand normal probabilities in terms of the graph (i.e. draw a normal curve, shade the area that you want). As the saying goes, "Garbage in, garbage out." If you have a fancy tool or gadget, you also need to know how to operate it before it will be of any use.