# Sample and Population Variance in R

Sample and Population Variance in R, The variance is a metric for determining how dispersed data values are around the mean.

Variance is the expectation of a random variable’s squared departure from its mean in probability theory and statistics, and it informally indicates how far a set of (random) values is spread out from its mean.

The formula for calculating a population’s variance is

`σ2 = Σ (xi – μ)2 / N`

where μ is the population mean, xi is the ith population element, N is the population size, and is basically Σ a fancy symbol for “sum.”

To determine a sample’s variance, use the following formula:

`s2 = Σ (xi – xbar)2 / (n-1)`

where xbar represents the sample mean, xi represents the sample’s ith element, and n represents the sample size.

## Calculate Sample & Population Variance in R

Assume we have the following R dataset and stored in data1.

Let’s create a data set values

`data1<- c(12,84, 5, 17, 18, 11, 13, 19, 69, 92,15,10,55)`

The var() function in R can be used to calculate sample variance.

Let’s calculate the sample variance

```var(data1)
957.8974```

The population variance can be calculated by multiplying the sample variance by (n-1)/n as follows.

Now we can calculate the length of the data1

```n <- length(data1)
n
13```

It’s ready to find population variance

```var(data1) * (n-1)/n
884.213```

It’s important to remember that the population variance is always lower than the sample variance.

In practice, we calculate sample variances for datasets because collecting data for a whole population is uncommon.

Calculate the Sample Variance of Multiple Columns as an example

Let’s say we have the following R data frame:

Now we can create a data frame

```data2 <- data.frame(X=c(12, 35, 55, 48, 54, 12, 8, 10),
Y=c(12, 24, 33, 77, 5, 46, 71, 106),
Z=c(1, 2, 63, 8, 12, 77, 92, 102))
data2```
```   X   Y   Z
1 12  12   1
2 35  24   2
3 55  33  63
4 48  77   8
5 54   5  12
6 12  46  77
7  8  71  92
8 10 106 102```

To determine the sample variance of each column in the data frame, we can use the sapply() function:

Yes, now based on sapply we can find each column’s sample variance.

`sapply(data2, var)`
```   X         Y         Z
439.6429 1238.7857 1863.9821```

We can also determine the sample standard deviation of each column using the following code, which is essentially the square root of the sample variance:

To find each column’s sample standard deviation

```sapply(data2, sd)
X        Y        Z
20.96766 35.19639 43.17386```

When it comes to data analysis, Sapply is a highly handy function.

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