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Normal distribution calculator

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Enter mean, standard deviation and cutoff points and this calculator will find the area under standard normal curve. The calculator will generate a step by step explanation along with the graphic representation of the probability you want to find.

Find the probability P( 80 < Z < 112)

solution

P(80 < X < 112) = 0.6678

explanation

Step 1: Sketch the curve. The probability that 80 < Z < 112 is equal to the blue area under the curve.

Step 2: Since $ \mu = 100 $ and $ \sigma = 16 $ we have:

$$ P~(~ 80 < X < 112 ~) = P~(~80 - \color{blue}{ 100 } < ~X - \color{blue}{\mu} < 112 - \color{blue}{ 100 } ~) = P~\left(~\frac{ 80 - 100}{ \color{blue}{ 16 } } < \frac{ X - \mu}{\color{blue}{\sigma}} < \frac{ 112 - 100}{ \color{blue}{ 16 } } \right) $$

Step 3: Since $ Z = \dfrac{x-\mu}{\sigma} $ , $ \dfrac{ 80 - 100}{ 16 } = -1.25$ and $ \dfrac{ 112 - 100}{ 16 } = 0.75$ we have:

$$ P~(~ 80< X < 112 ~) = P~(~ -1.25 < Z < 0.75 ~) $$

Step 4: Use the standard normal table to conclude that:

$$ P~(~ -1.25 < Z < 0.75 ~) = 0.6678 $$

Note: Visit Z - score calculator Z - score calculator for a step by step explanation on how to use the standard normal table.

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Normal distribution calculator
Find the area under normal distribution curve
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If X is a normally distributed variable with mean and standard deviation , find one of the following probabilities:

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Examples
ex 1:
A normally distributed random variable X has a mean of μ = 20 and a standard deviation of σ = 4. Determine the probability that a randomly selected x-value is between 15 and 22.
ex 2:
The final exam scores in a statistics class were normally distributed with a mean of μ = 58 and a standard deviation of σ = 4. Find the probability that a randomly selected student scored more than 62 on the exam.
ex 3:
The target inside diameter is μ = 50 mm but records show that the diameters follows a normal distribution with mean μ = 50 mm and standard deviation σ = 0.05 mm. An acceptable diameter is one within the range 49.9 mm to 50.1 mm. What proportion of the output is acceptable?
TUTORIAL

About this calculator

The normal distribution graph is also known as the bell curve. It is characterized by two parameters. The first one is the mean of a distribution; the graph is always symmetric about the mean, which means that half of the observations are greater than mean and half are lesser.

The second parameter of a normal distribution is the standard deviation, which determines the dispersion of data around the mean. If the standard deviation is large, then the data are more dispersed, and vice versa. 68% of observations are one standard deviation distant from the mean.

Normal distribution problems

Case 1: find an area above a certain value.

Example: The average height of 18-year-olds is 165 cm with a standard deviation of 8 cm. Find the percentage of 18-year-olds who are taller than 175 cm.

Graphical representation of an example 1

First we will find z-score

z = (x - μ)/σ = (175 - 165)/8 = 0.8

Note that P(X > 175) = P(Z > 1.25) = 1 - P(Z < 1.25)

Use standard normal table to get P(Z < 1.25) = 0.8944. So,

P(X > 175) = 1 - 0.8944 = 0.1056

Case 2: Find an area below a certain value.

Example: Rainfall in some area follows a normal distribution, with a mean of 250 mm and a standard deviation of 25. What is the probability that rainfall will be below 220 mm?

Graphical representation of an example 2

Like in the last example, we must first determine the z-score.

z = (x - μ)/σ = (220 - 250)/25 = -1.2

Now we use standard normal table to get:

P(X<220) = P(Z<-1.2) = 1 - P(Z<1.2) = 1 - 0.8849 = 0.1151

Case 3: Find an area between two values.

Example: IQ is normally distributed with a mean of 100 and a standard deviation of 16. What percentage of people have an IQ between 80 and 112.

Graphical representation of an example 3

In this example we need to find z-score for both cutoff point.

z = (x - μ)/σ = (80 - 100)/16 = -1.25

z = (x - μ)/σ = (112 - 100)/16 = 0.75

Now we use standard normal table to get:

P(80 < X <112) = P(-1.25 < Z< 0.75) = 0.6678

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