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## Statistics and probability

### Course: Statistics and probability > Unit 4

Lesson 5: Normal distributions and the empirical rule- Qualitative sense of normal distributions
- Normal distribution problems: Empirical rule
- Standard normal distribution and the empirical rule (from ck12.org)
- More empirical rule and z-score practice (from ck12.org)
- Empirical rule
- Normal distributions review

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# Normal distributions review

Normal distributions come up time and time again in statistics. A normal distribution has some interesting properties: it has a bell shape, the mean and median are equal, and 68% of the data falls within 1 standard deviation.

## What is a normal distribution?

Early statisticians noticed the same shape coming up over and over again in different distributions—so they named it the normal distribution.

Normal distributions have the following features:

- symmetric bell shape
- mean and median are equal; both located at the center of the distribution
of the data falls within$\approx 68\mathrm{\%}$ standard deviation of the mean$1$ of the data falls within$\approx 95\mathrm{\%}$ standard deviations of the mean$2$ of the data falls within$\approx 99.7\mathrm{\%}$ standard deviations of the mean$3$

*Want to learn more about what normal distributions are? Check out this video.*

### Drawing a normal distribution example

The trunk diameter of a certain variety of pine tree is normally distributed with a mean of $\mu =150{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ and a standard deviation of $\sigma =30{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ .

**Sketch a normal curve that describes this distribution.**

**Solution:**

**Step 1**: Sketch a normal curve.

**Step 2**: The mean of

**Step 3**: Each standard deviation is a distance of

### Finding percentages example

A certain variety of pine tree has a mean trunk diameter of $\mu =150{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ and a standard deviation of $\sigma =30{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ .

**Approximately what percent of these trees have a diameter greater than**$210{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ ?

**Solution**:

**Step 1:**Sketch a normal distribution with a mean of

**Step 2:**The diameter of

**Step 3:**Add the percentages in the shaded area:

**About**$2.5\mathrm{\%}$ of these trees have a diameter greater than $210{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}.$

*Want to see another example like this? Check out this video.*

*Want to practice more problems like this? Check out this exercise on the empirical rule.*

### Finding a whole count example

A certain variety of pine tree has a mean trunk diameter of $\mu =150{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ and a standard deviation of $\sigma =30{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ .

A certain section of a forest has $500$ of these trees.

**Approximately how many of these trees have a diameter smaller than**$120{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ ?

**Solution**:

**Step 1:**Sketch a normal distribution with a mean of

**Step 2:**The diameter of

**Step 3:**Add the percentages in the shaded area:

About $16\mathrm{\%}$ of these trees have a diameter smaller than $120{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}.$

**Step 4: Find how many trees in the forest that percent represents.**

We need to find how many trees $16\mathrm{\%}$ of $500$ is.

**About**$80$ trees have a diameter smaller than $120{\textstyle \phantom{\rule{0.167em}{0ex}}}\text{cm}$ .

## Want to join the conversation?

- Hello folks, For your finding percentages practice problem, the part of the explanation "the upper boundary of 210 is one standard deviation above the mean" probably should be two standard deviations. You do a great public service. Thanks.(25 votes)
- Nice one Richard, we can all trust you to keep the streets of Khan academy safe from errors(30 votes)

- Anyone else doing khan academy work at home because of corona?(20 votes)
- how exactly how you calculating the percentage of data lies within those standard deviations? like 1SD its 34%?(2 votes)
- Why do the mean, median and mode of the normal distribution coincide?(2 votes)
- 16% percent of 500, what does the 500 represent here? and where it was given in the shape(1 vote)
- 500 represent the number of total population of the trees. And the question is asking the NUMBER OF TREES rather than the percentage. So we need to figure out the number of trees that is 16 percent of the 500 trees, which would be 0.16*500.

Hope it helps.(2 votes)

- What is the mode of a normal distribution? The way I understand, the probability of a given point(exact location) in the normal curve is 0. So,is it possible to infer the mode from the distribution curve?(1 vote)
- hi. How do I find the standard deviation when I know that the distribution is approximately normal (n>25) and the mean is equal to 200?(0 votes)
- anyone know how to find Q3 using normal distribution with mean of 268 and standard deviation of 15.(1 vote)
- 1. Look up the standard normal table and find the z-

score that corresponds to the value 0.75

2. (x-268)/15 = z-score. Here x is the third quartile(1 vote)

- How do you find a specific percent within the deviations that isn't a sum of the existing groupings for percent? As an example, how would you find something like 85% of the data?(0 votes)
- There is standard normal table which is used for this purpose. This is exactly what is covered in the up coming modules.(1 vote)

- hello, I am really stuck with the below question, and unable to understand on text. I'd be really appreciated if someone can help to explain this quesion

6) The total home-game attendance for major-league baseball is the sum of all attendees for all stadiums during the entire season. The home attendance ( in millions) for a number of years is shown in the table below

Year Home Attendance (millions)

1978 40.6

1979 43.5

1980 43.0

1981 26.6

1982 44.6

1983 46.3

1984 48.7

1985 49.0

1986 50.5

1987 51.8

1988 53.2

a) Make a scatterplot showing the trend in home attendance. Describe what you see.

Inline image 2

a) Make a scatterplot showing the trend in home attendance. Describe what you see.

b) Determine the correlation, and comment on its significance.

c) Find the equation of the line of regression. Interpret the slope of the equation.

d) Use your model to predict the home attendance for 1998. How much confidence do you have in this

prediction? Explain.

e) Use the internet or other resource to find reasons for any outliers you observe in the scatterplot.(1 vote)