# How to use the STANDARDIZE function

**What is the STANDARDIZE function?**

The STANDARDIZE function calculates a normalized value from a distribution characterized by mean and standard_dev.

#### Table of Contents

## 1. Introduction

**What is a distribution?**

A distribution describes how data is distributed across possible values. It shows the frequency of values.

**What is the mean of a distribution****?**

The average value (arithmetic mean) of a distribution, calculated by summing numbers and dividing by the count.

The AVERAGE function calculates the arithmetic mean.

**What is the standard deviation of a distribution****?**

Standard deviation is a measure of dispersion that indicates how spread out the values in a dataset are from the mean. It is represented by the Greek letter sigma (σ).

The formula for calculating standard deviation is:

σ = √Σ(x - μ)2 / (N - 1)

Where:

σ = Standard deviation

Σ = Sum of

x = Values in the dataset

μ = Mean of the dataset

N = Number of values in the dataset

(N - 1) = Sample correction factor

The STDEV.P and STDEV.S functions calculate the standard deviation.

**What is normalizing a distribution?**

Normalizing a distribution means rescaling it to have a arithmetic mean of 0 (zero) and standard deviation of 1. See below on how STANDARDIZE function is calculated.

**What is a normal distribution?**

A symmetrical bell-shaped distribution where data clusters around the mean. Defined by its mean and standard deviation. The NORM.DIST function lets you create a bell-shaped distribution.

**What is a standard normal distribution?**

A standard normal distribution is a normal distribution with the mean of 0 (zero) and the standard deviation is 1.

**Why normalize a distribution?**

The shape of the normalized distribution allows you to determine

- the statistical method to use
- identify potential issues with the data (outliers, skewness, kurtosis etc)
- compare distributions across different groups or conditions

## 2. STANDARDIZE function Syntax

STANDARDIZE(*x*, *mean*, *standard_dev*)

## 3. STANDARDIZE function Arguments

x |
Required. The value you want to normalize. |

Mean |
Required. The arithmetic mean of the distribution. |

Standard_dev |
Required. The standard deviation of the distribution. |

## 4. STANDARDIZE Function Example 1

**A manufacturing company produces bolts with a target diameter of 10 mm. The process has a mean of 10.2 mm and a standard deviation of 0.3 mm. If a bolt has a diameter of 9.8 mm what is its standardized value?**

The arguments are:

- x = 9.8 mm
- mean = 10.2 mm
- standard_dev = 0.3 mm

These arguments are specified in cells C13,C14, and C15 respectively, in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 0 and the standard deviation is 1. The black lines represents the intersection between the standardized value and the blue curve which is the normal distribution.

Formula in cell C17:

The formula returns -1.333333 which represents the standardized x value, you can find that value on the chart above.

In the image above, locate the value 9.8 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the probability mass function for a normal distribution. Then, follow the point of intersection horizontally towards the y-axis to the left. You will find that the corresponding value on the y-axis is approximately 0.164.

You can use the PHI function to calculate the y value:

returns approx. 0.164

## 5. STANDARDIZE Function Example 2

**In a study of reaction times the mean response time was 250 milliseconds with a standard deviation of 30 milliseconds. If a participant had a reaction time of 260 milliseconds, what was their standardized reaction time?**

The arguments are:

- x = 260 ms
- mean = 250 ms
- standard_dev = 30 ms

These arguments are specified in cells C13,C14, and C15 respectively, in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 0 and the standard deviation is 1. The black lines represents the intersection between the standardized value and the blue curve which is the normal distribution.

Formula in cell C17:

The formula returns 0.333333 which represents the standardized x value, you can find that value on the chart above.

In the image above, locate the value 0.333333 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the probability mass function for a normal distribution. Then, follow the point of intersection horizontally towards the y-axis to the left. You will find that the corresponding value on the y-axis is approximately 0.377.

You can use the PHI function to calculate the y value:

returns approx. 0.377

## 6. STANDARDIZE Function Example 3

**In a class of 50 students the final exam scores were recorded. The mean score was 75 and the standard deviation was 10. If a student scored 85 on the exam what was their standardized score?**

The arguments are:

- x = 85
- mean = 75
- standard_dev = 10

These arguments are specified in cells C13,C14, and C15 respectively, in the image above.

The image above shows a chart containing a blue curve representing the probability mass function of a normal distribution where the mean is 0 and the standard deviation is 1. The black lines represents the intersection between the standardized value and the blue curve which is the normal distribution.

Formula in cell C17:

The formula returns 1 which represents the standardized x value, you can find that value on the chart above.

In the image above, locate the value 1 on the x-axis. From that point, draw an imaginary vertical line upwards until it intersects with the blue curve, which represents the probability mass function for a normal distribution. Then, follow the point of intersection horizontally towards the y-axis to the left. You will find that the corresponding value on the y-axis is approximately 0.24.

You can use the PHI function to calculate the y value:

returns approx. 0.24

## 7. STANDARDIZE function not working

STANDARDIZE returns a #NUM! error if argument *standard_dev* ≤ 0

## 8. How is the STANDARDIZE function calculated?

You can standardize any normal distribution like this:

z = (x - *µ)/σ*

*z = z-score
µ* is the mean.

*σ*is the standard deviation.

### Functions in 'Statistical' category

The STANDARDIZE function function is one of 73 functions in the 'Statistical' category.

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