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.

Nice one Richard, we can all trust you to keep the streets of Khan academy safe from errors

What is the empirical rule

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!

16% percent of 500, what does the 500 represent here? and where it was given in the shape

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.

Main content

Course: Statistics and probability > Unit 4

Lesson 5: Normal distributions and the empirical rule

Normal distributions review

Google Classroom

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
$\approx 68 %$ ‍ of the data falls within $1$ ‍ standard deviation of the mean
$\approx 95 %$ ‍ of the data falls within $2$ ‍ standard deviations of the mean
$\approx 99.7 %$ ‍ of the data falls within $3$ ‍ standard deviations of the mean

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

μ = 150 cm

and a standard deviation of

σ = 30 cm

Sketch a normal curve that describes this distribution.

Solution:

Step 1: Sketch a normal curve.

Step 2: The mean of

150 cm

goes in the middle.

Step 3: Each standard deviation is a distance of

30 cm

Practice problem 1

The heights of the same variety of pine tree are also normally distributed. The mean height is

μ = 33 m

and the standard deviation is

σ = 3 m

Which normal distribution below best summarizes the data?

Finding percentages example

A certain variety of pine tree has a mean trunk diameter of

μ = 150 cm

and a standard deviation of

σ = 30 cm

Approximately what percent of these trees have a diameter greater than $210 cm$ ‍?

Solution:

Step 1: Sketch a normal distribution with a mean of

μ = 150 cm

and a standard deviation of

σ = 30 cm

Step 2: The diameter of

210 cm

is two standard deviations above the mean. Shade above that point.

Step 3: Add the percentages in the shaded area:

2.35 % + 0.15 % = 2.5 %

About $2.5 %$ ‍ of these trees have a diameter greater than $210 cm .$ ‍

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

practice problem 2

Approximately what percent of these trees have a diameter between $90$ ‍ and $210$ ‍ centimeters?

%

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

μ = 150 cm

and a standard deviation of

σ = 30 cm

A certain section of a forest has

500

of these trees.

Approximately how many of these trees have a diameter smaller than $120 cm$ ‍?

Solution:

Step 1: Sketch a normal distribution with a mean of

μ = 150 cm

and a standard deviation of

σ = 30 cm

Step 2: The diameter of

120 cm

is one standard deviation below the mean. Shade below that point.

Step 3: Add the percentages in the shaded area:

0.15 % + 2.35 % + 13.5 % = 16 %

About

16 %

of these trees have a diameter smaller than

120 cm .

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

We need to find how many trees

16 %

500

is.

16 % of 500 = 0.16 \cdot 500 = 80

About $80$ ‍ trees have a diameter smaller than $120 cm$ ‍.

practice problem 3

A certain variety of pine tree has a mean trunk diameter of

μ = 150 cm

and a standard deviation of

σ = 30 cm

A certain section of a forest has

500

of these trees.

Approximately how many of these trees have a diameter between $120$ ‍ and $180$ ‍ centimeters?

\approx

trees

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Richard
Posted 8 years ago. Direct link to Richard's post “Hello folks, For your fi...”
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.
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(27 votes)
Answer
- Dorian Bassin
  Posted 6 years ago. Direct link to Dorian Bassin's post “Nice one Richard, we can ...”
  Nice one Richard, we can all trust you to keep the streets of Khan academy safe from errors
  Comment on Dorian Bassin's post “Nice one Richard, we can ...”
  (31 votes)
Admiral Snackbar
Posted 4 years ago. Direct link to Admiral Snackbar's post “Anyone else doing khan ac...”
Anyone else doing khan academy work at home because of corona?
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29vsingharia
Posted 3 months ago. Direct link to 29vsingharia's post “What is the empirical rul...”
What is the empirical rule
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(2 votes)
Answer
- Ian Pulizzotto
  Posted 2 months ago. Direct link to Ian Pulizzotto's post “For a normal distribution...”
  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!
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  (3 votes)
Rupendra Choudhary
Posted a year ago. Direct link to Rupendra Choudhary's post “how exactly how you calcu...”
how exactly how you calculating the percentage of data lies within those standard deviations? like 1SD its 34%?
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(2 votes)
Answer
- daniella
  Posted 4 months ago. Direct link to daniella's post “The percentages for the d...”
  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.
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  (1 vote)
Chowdhury Amir Abdullah
Posted 7 years ago. Direct link to Chowdhury Amir Abdullah's post “Why do the mean, median a...”
Why do the mean, median and mode of the normal distribution coincide?
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(2 votes)
Answer
- Luis Fernando Hoyos Cogollo
  Posted 6 years ago. Direct link to Luis Fernando Hoyos Cogollo's post “Watch this video please h...”
  Watch this video please https://www.khanacademy.org/math/statistics-probability/modeling-distributions-of-data/modal/v/median-mean-and-skew-from-density-curves
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  (1 vote)
Alobaide Sinan
Posted 3 years ago. Direct link to Alobaide Sinan's post “16% percent of 500, what ...”
16% percent of 500, what does the 500 represent here? and where it was given in the shape
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(1 vote)
Answer
- lily.
  Posted 2 years ago. Direct link to lily.'s post “500 represent the number ...”
  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.
  Button navigates to signup page
  (2 votes)
Rohan Suri
Posted 4 years ago. Direct link to Rohan Suri's post “What is the mode of a nor...”
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?
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(1 vote)
Answer
- daniella
  Posted 4 months ago. Direct link to daniella's post “The mode of a normal dist...”
  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.
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  (1 vote)
alnreddy17
Posted 5 months ago. Direct link to alnreddy17's post “someties they are taking ...”
someties they are taking 99.7% and sometimes 100%
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(1 vote)
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Kasia
Posted 3 years ago. Direct link to Kasia's post “hi. How do I find the sta...”
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?
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Answer
- daniella
  Posted 4 months ago. Direct link to daniella's post “If you know that the dist...”
  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.
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b3auty93
Posted 7 years ago. Direct link to b3auty93's post “anyone know how to find Q...”
anyone know how to find Q3 using normal distribution with mean of 268 and standard deviation of 15.
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(1 vote)
Answer
- HIKIKOMORI
  Posted 3 years ago. Direct link to HIKIKOMORI's post “1. Look up the standard n...”
  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
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  (1 vote)