# What does the standard deviation actually tell me

## Standard Deviation Interpretation in Detail: What is the Meaning of Standard Deviation?

One of the most important indicators for statistical analyzes is certainly the standard deviation (abbreviation standard deviation: SD or s). Numerous statistical methods ultimately result in an analysis of the standard deviation or the variance (squared standard deviation). The analysis of variance, for example, already has the variance in its name. But the standard deviation also forms the fundamental building block of the analysis for many other methods. A basic understanding of standard deviation is therefore essential for anyone dealing with data analysis results. In this article we would therefore like to present you with an easily understandable introduction to the standard deviation interpretation.

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### The following questions are answered in this article

- What does standard deviation mean?
- How do you go about calculating the standard deviation and variance?
- How can I meaningfully interpret the standard deviation?

### Standard deviation meaning - a practical example

Basically, the standard deviation and the variance indicate how much the data deviates from the mean. In other words, the standard deviation expresses how much the data points differ from one another. An example to illustrate this:

A company would like to know how satisfied its customers are with the product they have purchased. In a customer survey, customers answer the following question, among other things: "What is the probability that you would recommend our product?"

The standard deviation also plays a role in so-called NPS surveys

### What if the standard deviation is zero?

Let's assume that all of the customers surveyed gave this question a value of 7. In this case the average would be 7 and there would be no difference at all between the data points; the standard deviation would be 0. In such a case there would actually be no approach for a classical statistical analysis: One of the most important goals of statistics is to explain and predict differences between data points. If all customers are the same, there are no differences to explain or predict.

A distribution with a standard deviation of 0 would be very unusual

Such a case is of course extremely unlikely. Even if all customers would normally rate the service with a 7, there are practically always measurement errors. Some customers may rate the product a little higher because you had a good day. Others rate the product a little lower because you are generally difficult to please. Another customer just clicked, and so on.

### Standard deviation with approximately normal distribution

In this case, the data will be around the value 7, so the average would be 7. However, all values will not be identical, so some values will be below 7 or above 7. If the data is roughly normally distributed, most of the values will be around the mean. Values with a small deviation from the mean will occur more frequently than values with a very large deviation.

Example of an approximately normal distribution

If we now want to know how much the values differ from one another, the standard deviation is an extremely practical quantity. For the standard deviation, we first need to calculate the variance.

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### From the variance to the standard deviation

Variance is the statistical expression for the spread of the data. The variance indicates how far the data differ on average from the mean. The greater the variance, the further the data are from the mean.

Example of high variance

Example of low variance

We can simply express the deviation from the mean as:

Deviation = x- \ bar {x}

Where \ bar {x} represents the mean value.

If the value is now smaller than the average, the deviation is negative. However, we would like a high deviation to be expressed by a high value. The squared deviations are therefore used to define the variance.

Squared deviation = \ left (x- \ bar {x} \ right) ^ {2}

The population variance is defined as the mean of all the squared deviations:

Var \ left (x \ right) = \ frac {\ sum _ {i = 1} ^ {n} \ left (x_ {i} - \ bar {x} \ right) ^ {2}} {n}

This means that we first calculate the squared deviation for all values and then form the average from this. Estimating the variance with a small sample, however, underestimates this. To correct this, s^{2} usually through *n - 1* divided:

s ^ {2} = \ frac {\ sum _ {i = 1} ^ {n} \ left (x_ {i} - \ bar {x} \ right) ^ {2}} {n-1}

The standard deviation (abbreviation standard deviation: *SD* or *s*) is simply defined as the square root of the variance:

s = \ sqrt [] {s ^ {2}} = \ sqrt [] {\ frac {\ sum _ {i = 1} ^ {n} \ left (x_ {i} - \ bar {x} \ right) ^ {2}} {n-1}}

What does this value say now? As we will show you in the next section, the standard deviation has some very useful properties. These make the standard deviation interpretation very easy.

### Interpretation of standard deviation: practical rules of thumb

If the data is in a normal distribution, you can read a lot of useful information from a standard deviation interpretation. If the data are almost normally distributed, about 68% of all data are within one standard deviation of the mean. About 95% are within 2 standard deviations (more precisely: 1.96) and 99.7% are within 3 standard deviations. This is also known as the 68-95-99.7 rule. In the case of normally distributed data, a quick look at the mean value and the standard deviation is sufficient to get an idea of the range in which most of the data are located.

A confidence interval can also be quickly calculated using the standard deviation and the mean. A 95% confidence interval has a critical z-value of 1.96. The mean value is therefore with 95% certainty between M-1.96 \ ast \ frac {SD} {\ sqrt [] {n}} and M + 1.96 \ ast \ frac {SD} {\ sqrt [] { n}}.

### Standard deviation interpretation in practice

Back to our example: An initial analysis of the customer survey showed that the data is almost normally distributed. A descriptive analysis resulted in the following key figures:

SPSS output with standard deviation

From the mean value and the standard deviation, the 68-95-99.7 rule can be used to determine how the data should roughly be distributed with a normal distribution (abbreviation mean: M, abbreviation standard deviation: SD).

Share of data | Rule of thumb | Approximate area |

68% | M +/- SD | 5,60 – 8,24 |

95% | M +/- (SD * 2) | 4,27 – 9,57 |

99,7% | M +/- (SD * 3) | 2,95 – 10 |

In the table above, note that 99.7% of the data should actually be between 2.95 - 10.89, but 10 represents the maximum possible value of the scale.

In order to be able to estimate more precisely in which area customer satisfaction is likely to be, a 95% confidence interval should be determined. This can of course be done easily in SPSS, but it can also be done quickly by hand:

95 \% KI = M \ pm 1.96 \ ast \ frac {SD} {\ sqrt [] {n}} = 6.92 \ pm 1.96 \ ast \ frac {1.32} {\ sqrt [] {100}} = 6.92 \ pm 0.26 = \ left [6.66; 7.18 \ right]

The mean value is between 6.7 and 7.2 with 95% certainty. Actually, an average value of 8 is aimed for. A subsequent study should clarify which factor could contribute to greater satisfaction. This could be done using an ANOVA or regression analysis.

### Summary: Interpretation of standard deviation made easy

In this article we have given you a brief introduction to standard deviation interpretation. The standard deviation is a key figure in statistics and forms the basic building block for many important statistical methods. But just in combination with the mean, the standard deviation meaning is already very informative, especially with normally distributed data.

Following an initial analysis of the mean and standard deviation, an in-depth analysis of the data with advanced methods is often a good idea. If you would like advice on how to proceed, our Novustat experts will be happy to assist you.

### Further sources:

Wissenschafts-Tower: Basics of the standard deviation

Stanford: The Normal Distribution

Explanation of the difference between standard deviation and standard error

Keywords: anova, descriptive statistics, spss evaluation, standard deviation interpretation, analysis of variance

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