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- Normal Distribution Calculator

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.

**solution**

**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:

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

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

**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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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?

Related calculators

TUTORIAL

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.

**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.

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?

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.

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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