# Range, variance, and standard deviation

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## Biostatistics and epidemiology

#### Biostatistics

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### Range, variance, and standard deviation

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A research study of 100 patients shows that their calcium levels range from 8.8-15.1 mg/dL, with a mean of 12.1 mg/dL. The calcium levels fall in a normal distribution, with a standard deviation of 1.0 mg/dL. Based on this study, we know that the percentage of calcium values below 10.1 is approximately:

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#### Content Reviewers:

Rishi Desai, MD, MPH, Yifan Xiao, MD#### Contributors:

Evan Debevec-McKenney, Anju PaulTo understand a set of data - having a single number like the mean or median gives us a one number summary, but understanding how the data is distributed is also very important - and that’s where the range, variance, and standard deviation can be helpful.

For example, let’s say we are looking at the weight of 10 people and we divided them into groups A and B.

```
weight of group A(in kg) weight of group B(in kg)
40 45 50 55 60 10 30 50 70 90 .
```

Mean = (40+45+50+55+60)/5= 50kg Mean = (10+30+50+70+90)/5= 50kg

Now, if you calculate the mean weight of group A and group B, you will find both of them have the same value of 50 kg, but the weights of individuals in group A are much more centered around the mean than in group B.

So let’s start by looking at the range, which is the difference between the highest and lowest value in a dataset.

In group A, we have (60-40)=20 kg, whereas in group B we have (90-10)=80 kg.

So far so good. But now, let’s say we have decided to include another group called group C.

Weight of Group C (in kg) 10 45 50 55 90. Mean = (10+45+50+55+90)/5= 50 kg Range 90-10 = 80kg

So even when we change two data points, group C still has the same mean and range as group B since it depends only on the highest and lowest values, thus it provides no information about how the rest of the data points are distributed.

In this situation, it’s clear that we need a better idea of how all of the values are distributed and to do that we can look at the variance.

To calculate the variance, which is written out as σ2, we take each data point (x), subtract it from the mean (x-bar), and then we square this value so we don’t end up with a negative number.

Next, we add up the squared values and divide that result with the total number of data points (n).

So, let’s use this formula to calculate the variance for group A, B, and C where all three had a mean of 50.

So, for group A, we get: (40-50)2+(45-50)2+(50-50)2+(55-50)2+(60-50)2/(5)=50 kg2

For group B, we get: (10-50)2+(30-50)2+(50-50)2+(70-50)2+(90-50)2 /(5) = 800 kg2