### Descriptive Statistics (Animal Science)

##### Mean

Most of us are familiar with the mean or “average” as summing all the data and dividing by the total number of data, but if you’re interested, the statisticians represent that calculation in a formula like this:

If you have a sample of data consisting of $n$ observations $(x_1, \dots, x_n)$ then the mean $(\bar{x})$ of this sample is calculated using the formula:

$$\bar {x} = \frac{1}{n}\sum\limits_{i=1}^{n}\ x_i.$$

##### Population Mean

The population mean, denoted by $\mu$ (the Greek letter 'mu'), is the average for the entire population i.e. it would be the sample mean if $n$ was equal to size of the whole population. For example, if we were measuring the wing span of albatrosses, there would exist a population mean for this particular measurement. We would know this if we took all the albatrosses in the world, measured their wing spans and took the average. Usually we do not know the population mean, which is why we draw conclusions from the sample mean.

##### Mode

The mode is the most frequently occurring value in the data set. For instance, in the set of data for hamster litter: $3,4,4,5,4,6,7,8$ the modal value of litter sizes is $4$ as $4$ occurs most frequently.

##### Median

The median, simply put, is the “middle” value for a set of data. To calculate this value, order the data values by size, and the value that is in the middle is the median. For example in the ordered set of nine numbers $1,3,4,4,6,7,8,8,9$ the median is the fifth number in this set which is $6$.

See Mean, median and mode and Weighted averages for more information and more in-depth examples.
##### Range

The range measures the difference between size of the largest value in the data set to the smallest. We calculate this using the formula:

$$\text{Range} =\text{Largest value - smallest value}.$$

##### Sample Standard deviation

The standard deviation (SD) is a measurement that calculates how much the data deviates from the mean and is usually denoted by $\sigma_{n-1}$ or $S$. The standard deviation is the positive square root of the variance. It is calculated using the formula:

$$\sigma_{n-1} = \sqrt{\frac{\sum\limits_{i=1}^n(x_i - \bar {x})^2}{n-1}}.$$

##### Sample Variance

The variance is the measurement of the squared differences of the data values from the mean (square of the standard deviation). We square them because values less than the mean would have a negative difference and cancel out values greater than the mean. Squaring the differences ensures every value is positive. The variance is calculated by the formula:

$$\text{variance} = \frac{1}{n-1}\sum\limits_{i=1}^n(x_i - \bar {x})^2.$$

##### Standard Error (SE) of the Mean

The sample mean is an estimator of the population mean. The standard error is how much the sample mean deviates from the population mean. We calculate it using the following formula:

$$\text{Standard Error (SE)} = \frac{\sigma_{(n-1)}}{\sqrt{n}}.$$

Notation in the above formulas explained:

\begin{align} \bar{x} &= \text{The sample mean} \\ x &= \text {An individual data value} \\ \sigma_{n-1} \text{ or S} &= \text{Sample standard deviation} \\ n &= \text{Sample size (number of observations)} \\ \sum \limits_{i=1}^n{x_i} &= \text{Sum of of the data values} (x_1,\ldots,x_n). \end{align}

You can calculate the mean, variance and standard deviation on your calculator! See using your calculator.