# How to use the CONFIDENCE.NORM function

**What is the CONFIDENCE.NORM function?**

The CONFIDENCE.NORM function calculates the confidence interval for a population mean. In other words it calculates the margin of error, if the mean is μ then the confidence interval is μ ± CONFIDENCE.NORM.

Use the CONFIDENCE.NORM function when you have more than 30 observations. The CONFIDENCE.T function is for less than 30 observations, it accounts for the increased uncertainty associated with small sample sizes.

#### Table of Contents

## 1. Introduction

**What is a confidence interval?**

A confidence interval provides a range of plausible values for an unknown population parameter like a mean, based on a sample estimate and how much variability exists around that estimate. It is indicating the reliability of the estimate at a specified confidence level.

**What is a population mean?**

A population mean commonly represented by the Greek letter mμ is the arithmetic average value in a statistical population. It is estimated using the sample mean taken from a representative random sample of the entire population.

**What is the variability?**

Variability refers to the dispersion or spread of values in a data set and their tendency to deviate from the central mean value. It is quantified by statistical measures like variance, standard deviation, and interquartile range.

**What is the arithmetic average value in a statistical population?**

The arithmetic average value in a statistical population is the population mean, commonly represented by the Greek letter mu. It is equal to the sum of all the individual values divided by the total number of elements in the full population.

**What is the difference between the CONFIDENCE.T function and the CONFIDENCE.NORM function?**

The CONFIDENCE.NORM function calculates the confidence interval for a population mean using the normal distribution whereas the CONFIDENCE.T function calculates the confidence range for a population mean using a Student's t distribution.

CONFIDENCE.NORM(*alpha,standard_dev,size*)

CONFIDENCE.T(*alpha,standard_dev,size*)

## 2. CONFIDENCE.NORM Function Syntax

CONFIDENCE.NORM(*alpha,standard_dev,size*)

## 3. CONFIDENCE.NORM Function Arguments

alpha |
Required. The significance level. |

standard_dev |
Required. The standard deviation. |

size |
Required. The number of samples. |

**What is the significance level?**

The significance level, denoted α (alpha) , is the probability of mistakenly rejecting the null hypothesis when it is in fact true. It is providing a threshold used to judge whether sample evidence is sufficient to indicate a real effect or relationship.

**What is standard deviation?**

Standard deviation measures the amount of variation or dispersion of values in a dataset relative to the mean. It is calculated as the square root of the variance by taking

- each data point's deviation from the mean
- squaring it
- averaging the squares
- finally square rooting the result

STDEV.P function in Excel calculates the standard deviation.

## 4. CONFIDENCE.NORM Function Example 1

**You study plants, you have 70 plants in your inventory. The average height is 20 cm, and the standard deviation σ is 3 cm. How to estimate the mean μ height with a 95% confidence level?**

The CONFIDENCE.NORM function lets you calculate the interval that satisfies a given a significance level, standard deviation and a number representing the population size.

The image above shows the argument:

- alpha in cell C19 which is 1-0.95 = 0.05
- standard_dev in cell C20 = 3
- size in cell C21 = 70

Formula in cell C23:

The CONFIDENCE.NORM function returns 0.702781508276192 for the specified arguments above.We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:

The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:

In other words, there is a 95% probability that the mean μ is between approx. 19.297 and approx. 20.703

## 5. CONFIDENCE.NORM Function Example 2

**You work as an engineer at a company manufacturing electronic equipment. You have made 40 temperature measurements from one of the machines.**

**The measurements are in cell range E19:H28 displayed in the image above. The table below shows the same values:**

66 | 59 | 58 | 67 | 59 | 57 | 71 | 56 | 56 | 72 |

62 | 57 | 71 | 68 | 59 | 66 | 72 | 56 | 55 | 63 |

56 | 75 | 68 | 71 | 74 | 56 | 56 | 64 | 60 | 71 |

65 | 67 | 58 | 63 | 71 | 64 | 60 | 60 | 70 | 73 |

**How to estimate the mean μ temperature interval with a 80% confidence level?**

The CONFIDENCE.NORM function requires three arguments, the alpha, standard deviation and the size. The alpha is specified in question, 1 - 0.8 = 0.2

We can calculate the mean μ using the AVERAGE function. Formula in cell C24:

The average is calculated by adding all the measurements and then divide by the number of observations. The standard deviation is also easy to calculate in Excel. Formula in cell C25:

Here is how the STDEV.P function calculates the standard deviation:

We now have all the arguments needed to calculate the interval that satisfies a given significance level at 80% (0.8).

- alpha in cell C19 which is 1-0.8 = 0.2
- standard_dev in cell C20 approx. equal to 6.19
- size in cell C21 = 40

Formula in cell C23:

The CONFIDENCE.NORM function returns 1.25418662730462 for the specified arguments above. We can now calculate the upper and lower limit by adding the confidence interval from cell C23 to the mean μ. I have done this in cell C25:

The lower limit is calculated by subtracting the confidence level to the mean μ. Cell 26:

In other words, there is a 80% probability that the mean μ is between approx. 62.55 and approx. 65.05

## 6. CONFIDENCE.NORM Function Example 3

This example builds on the first example in section 4.

**A year has gone since you made your last calculations. You now have 100 plants in your inventory. The average height is 20 cm, and the standard deviation σ is 3 cm. How to estimate the mean μ height with a 95% confidence level?**

The only thing that has changed is the number of plants (size). Lets see if that changes the confidence interval.

The image above shows the argument, only one has changed and that one is in cell C21:

- alpha in cell C19 which is 1-0.95 = 0.05
- standard_dev in cell C20 = 3
- size in cell C21 = 100

Formula in cell C23:

The CONFIDENCE.NORM function returns approx. 0.588 for the specified arguments above. The interval is smaller which is understandable since the sample size has grown to 100.

There is a 95% probability that the mean μ is between approx. 19.41 and approx. 20.59. The interval is smaller which makes sense considering the size is larger.

## 7. CONFIDENCE.NORM function not working

The CONFIDENCE.NORM function returns:

- #VALUE! error value if any argument is non-numeric.
- #NUM! error value if
*alpha*<= 0 (zero) or*alpha*>= 1.*standard_dev*<= 0.*size*< 1.

CONFIDENCE.NORM function truncates *size* numbers to integers.

### Functions in 'Statistical' category

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

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