Standard deviations are scores around the mean of a distribution. It measures how much a set of scores is dispersed around an average measure of variability.

Deviations around the mean can be calculated to express it as a variance or a standard deviation.

## Standard Deviation

A **standard deviation** is a statistic that is calculated as the square root of a variance, or a data set calculated by taking the mean of the squared differences between each value and the mean value. Because the differences are squared, units of variance are not units of data. This is why a standard deviation is the square root of the variance. The points or units plotted from the variance becomes the data set.

Standard deviations and variances are common measures of dispersion.

σ = standard deviation |
σ ² = variance |

## Average/Norm

A standard deviation is how a set of data is plotted around the mean (average) of a set of data. It is how a data set compares to a calculated norm or standard. The further the data spreads from the mean, the greater the deviation from the norm. Likewise, the closer the data are to the mean, the closer the data are to the norm, which creates a steep curve. The standard deviation of a data set that equals zero indicates all values in the set are the same.

## Bell Curve

Standard deviation data sets are plotted and dispersed around a **bell curve**. A bell curve is a symmetrical curve that represents the distribution, values, and frequencies in a set of data. From the middle point at the top, it slopes in a downward arc on both sides. The top middle point is the mean value or the maximum probability.

Probability is likelihood of occurrence.

## Distribution

As the probability decreases, the slope of the bell curve falls away from the mean. This explains why a wide deviation indicates a greater deviation from the mean. A **normal distribution** of data means that the numbers in the standard deviation’s data set are close to the mean.Units are plotted on the x-axis, or the horizontal line, in relation to how this information aligns with the frequency for each value on the y-axis, or the vertical line.

For example, on the y-axis, salary, range may be from $50,000 to $200,000, with an average salary of $200,000 (top of the bell curve). Let’s say we plot the salaries of 30 people, or units, on the x-axis. The x-axis would reflect where each worker’s salary would be plotted according to how much each person made.

To the right of the median average, the worker would make more than $200,000, and to the left of the median average, he or she would make less than $200,000.

## Deviations

One standard deviation from the mean (red area) on the x-axis usually accounts for about 68% of the data set. Two standard deviations from the mean (green area) usually accounts for about 95% of the data set. Likewise, three standard deviations from the mean (blue area) usually accounts for about 99% of the data set. Lastly, four deviations from the mean would be the white area.

Using the example given, the further from the average salary of $200,000 earned (slope top), the greater the deviation from the mean (average). Also notice that the number of data sets would decrease with each standard deviation. Again, large standard deviation could reveal that not too many people are averaging $200,000 in earnings. A small standard deviation may be revealing that there are many who are averaging earnings around the norm for the 30 people plotted. Depending upon the size of the bell curve, another interpretation could be that everyone averages around the norm (a steep curve) or that not too many people are averaging $200,000 (a flatter slope).

## Lesson Summary

A **standard deviation** is a statistic that is calculated as the square root of a variance, or a data set calculated by taking the mean of the squared differences between each value and the mean value. Because the differences are squared, units of variance are not units of data. This is why a standard deviation is the square root of the variance. The points or units plotted from the variance become the data set. Standard deviations and variances are common measures of dispersion.A standard deviation is how a set of data is plotted around the mean (average) of a set of data.

It is how a data set compares to a calculated norm or standard. Standard deviation data sets are plotted and dispersed around a **bell curve**, which is a symmetrical curve that represents the distribution, values, and frequencies in a set of data. A **normal distribution** of data means that the numbers in the standard deviation’s data set are close to the mean.

One standard deviation from the mean (red area) on the x-axis usually accounts for about 68% of the data set. Two standard deviations from the mean (green area) usually accounts for about 95% of the data set. Likewise, three standard deviations from the mean (blue area) usually accounts for about 99% of the data set. Lastly, four deviations from the mean would be the white area.

## Learning Outcomes

Knowing the contents of this lesson could enable you to do the following:

- Define standard deviation, bell curve and normal distribution
- Explain why a standard deviation is the square root of the variance
- Plot and interpret standard deviation around a bell curve