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## Integrated math 3

### Course: Integrated math 3>Unit 12

Lesson 1: Normal distributions

# Basic normal calculations

Many measurements fit a special distribution called the normal distribution. In a normal distribution,
• $\approx 68\mathrm{%}$ of the data falls within $1$ standard deviation of the mean
• $\approx 95\mathrm{%}$ of the data falls within $2$ standard deviations of the mean
• $\approx 99.7\mathrm{%}$ of the data falls within $3$ standard deviations of the mean
Problem 1
A large sample of females had their systolic blood pressure measured. The mean blood pressure was $125$ millimeters of mercury and the standard deviation was $10$ millimeters of mercury.
Which normal distribution below best summarizes the data?

Problem 2
What percent of females had blood pressures between $105$ and $135$ millimeters of mercury?
$\mathrm{%}$

Problem 3
The sample had a total of $400$ females that participated.
About how many females in the sample had blood pressures higher than $145$ millimeters of mercury?
$\approx$
females

## Want to join the conversation?

• What is the Gauss curve have to do with this?
• For the empirical rule to be valid the data must be normally distributed, so the rules for percentages in the problems above would not hold true if the data didn't follow a gaussian or normal distribution.
• I answered the last question in the following way. I figured out the z score which was 0.9772 for value that is two standard deviations above normal. Therefore one can deduce that the sample size that lies to the right of this should be 0.0228 (1-0.9772). That multiplied by 400 gives me 9.1 approximated to 9. What is wrong with this method?
• I agree that using the z-Score is the most accurate answer. The answer given is using approximations of the normal and are not as accurate.
• For the titles of Articles, Videos, and Skills; should the wording follow the same grammar rules as the title of a book does?
e.g.: Should the titles be capitalized with the exception of conjunctions like and/or/but/ect?

I'm slightly confused, because KA only capitalizes the first word, but I can't find anywhere on the web where it tells you how to properly right the title.

Any help?

I'm slightly confused...
• Khan Academy is an exception to the usually title rules. Sometimes websites might only capitalize the first word or not follow the exact rules as in a book for some videos and articles. You are correct though, that rule is used commonly in books, movies, etc.
• where does the 13.5% come from in question 3?
• Empirical rule!
Ok, so 95% of the observations are within 2 standard deviations of the mean, and 68% of the observations are within 1 standard deviation of the mean, right?
Let's find the difference between 2 s.d. and 1 s.d. It will be 95%-68%=27%. But you have to divide this 27% by 2 because you have to find the percentage between 105 and 115 millimeters. There comes 13.5%!
• what if they ask you to solve for standard deviation with only knowing the range
• This question is not really meaningful for a normal distribution, since all normal distributions have infinite range.

For general data sets, knowing the range of a data set is not sufficient for finding its standard deviation. For example, the data sets 1,5,5,9 and 1,2,8,9 both have range 9-1=8, but 1,2,8,9 has the larger standard deviation because the values are spread out farther from the mean (5).

Have a blessed, wonderful day!
• last question was tricky, how u do it
• You multiply 400 by the percent of females that had blood pressure in that area, which is 2.5%. So, 400 x 2.5% (400 x 0.025 in decimal) gets you the answer, which is 10.