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### 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.(27 votes)
- Nice one Richard, we can all trust you to keep the streets of Khan academy safe from errors(31 votes)

- Anyone else doing khan academy work at home because of corona?(22 votes)
- What is the empirical rule(2 votes)
- For a normal distribution, about 68% of the data are within 1 standard deviation from the mean, about 95% of the data are within two standard deviations of the mean, and about 99.7% of the data are within three standard deviations of the mean.

Have a good day!(3 votes)

- how exactly how you calculating the percentage of data lies within those standard deviations? like 1SD its 34%?(2 votes)
- The percentages for the data within certain standard deviations in a normal distribution are derived from the properties of the normal distribution curve and the empirical rule. For example, the 34% for one standard deviation comes from the fact that the area under the normal curve within one standard deviation of the mean is symmetrically distributed around the mean and covers approximately 34% of the total area under the curve. This is a characteristic property of the normal distribution and is used as a rule of thumb to understand the distribution of data.(1 vote)

- 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)
- The mode of a normal distribution is the value at which the curve reaches its peak, which coincides with the mean and median in a normal distribution. While the probability of a specific point in a continuous distribution being exactly equal to a particular value is indeed 0, the mode is still a meaningful concept because it represents the most frequently occurring value in the distribution, even if it's not a precise point on the curve.(1 vote)

- someties they are taking 99.7% and sometimes 100%(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)
- If you know that the distribution is approximately normal and the sample size is sufficiently large (n > 25), you can estimate the standard deviation using the sample standard deviation formula, which involves calculating the square root of the variance. Alternatively, if you have access to the entire population data, you can calculate the population standard deviation directly.(1 vote)

- 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)