Standard deviation is a fundamental concept in statistics that quantifies the amount of variation in a set of data values. A low standard deviation indicates that the data points are closely packed around the mean (average), whereas a high standard deviation indicates that the data points more spread out. It is a critical tool in fields such as statistics, physics, psychology, and many more, providing insights into the reliability and predictability of data. This article will provide an in-depth account of this type of measure of variability.
Inhaltsverzeichnis
The Standard Deviation in Statistics– In a Nutshell
- Standard deviation is used to determine the variability in a dataset.
- You can calculate standard deviation using a formula or software.
- Another commonly used measure of variability is the mean absolute deviation (MAD).
Definition: Standard deviation
Standard deviation is a statistical measure that represents how much the values in a dataset deviate from the arithmetic mean. In other words, it quantifies the degree to which each data point deviates from the mean, or average, of the data set. A smaller standard deviation implies that the data points are closely clustered around the mean, while a larger standard deviation shows that the data points are more spread out.
The meaning of standard deviation
This statistic is used to measure the dispersion in a dataset. It shows you the average amount of variability or how far each value lies from the mean on average. This statistic is used with continuous data and shouldn’t be used with categorical data. Furthermore, it has to be used with datasets that have a normal distribution. Some good examples are height, temperature, and length.
Example
The scores in a given test were recorded as:
Scores: 20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%
The standard deviation in this sample is 24.5.
Note: A low deviation means that the data points are closer to the mean. A high standard deviation indicates that there is a wider range of values in the dataset. This particular example has a low deviation.
The empirical rule
The empirical rule is also known as the 68-95-99.7 rule. It works as a guide on how data is distributed in a normal distribution.
According to this rule, about 68% of the data points will fall within one standard deviation of the mean, and 95% of the data points will fall within two standard deviations of the mean.
The rule also states that 99.7% of the data points will fall within three standard deviations of the mean. You should use this rule to forecast future outcomes.
We can refer to our example above. According to the empirical rule, the following facts hold true:
Example
- 68% of scores are between 30.5% and 79.5%.
- 95% of scores are between 6% and 104%.
- 99.7% of scores are between -18.5% and 128.5%.
With the empirical rule, you can easily check for outliers in a normal distribution.
Standard deviation formulas
Data can be derived from a sample or population.
A population refers to the entire group that you intend to draw conclusions about.
On the other hand, a sample is a small group that is used for data collection. The formulas for this statistic are different for population and sample data.
Populations
Regarding calculating the standard deviation of a population, the following formulas are used:
| N | Number of values in the population |
| Σ | Sum of N |
| X | Individual values in the population |
| μ | Mean |
| σ | Population standard deviation |
| √ | Square root |
| s | Standard deviation for the sample |
| √ | Square root |
| x | Each value in the sample |
Calculating the standard deviation
There are some programs you can use to calculate the standard deviation automatically. If you want to calculate the standard deviation manually, you can follow these steps. We’ll use the dataset above to demonstrate this formula.
Example
20%, 30%, 40%, 50%, 60%, 70%, 80%, 90%.
Step 1: Find the mean
You have to start by finding the mean. This is done by adding up all values and dividing the sum by the sample size.
Example
- The sum of our values is 440.
- This has to be divided by 8 to get the mean.
- This will give you a mean of 55.
Step 2: Finding deviations from the mean
Next, you should find out each value’s deviation from the mean.
In our example, the deviations are as follows:
Example
| 20 | -35 |
| 30 | -25 |
| 40 | -15 |
| 50 | -5 |
| 60 | 5 |
| 70 | 15 |
| 80 | 25 |
| 90 | 35 |
Step 3: Square the deviations from the mean
You can then square the deviations from the mean:
Example
| -35 | 1225 |
| -25 | 625 |
| -15 | 225 |
| -5 | 25 |
| 15 | 225 |
| 25 | 625 |
| 35 | 1225 |
Step 4: Sum the squares
In this step, you have to find the sum of the squares.
Example
They add up to a total of 4175.
Step 5: Find the variance
You then have to find the variance. You can do this by dividing the sum of squares by (n-1). If you are dealing with a population instead of a sample, you can divide the sum of squares by N.
Example
Since we are using a sample, we will divide 4175 by (8-1). This gives you 596.428.
Step 6: Finding the square root of the variance
Finally, you will have to find the square root of the variance.
Example
Since the variance in our example is 596.428, our standard deviation will be 24.5. This means that, on average, the values in the dataset deviate from the mean by 24.5.
Standard deviation or other methods of variability
Standard deviation is only one way of measuring variability. You can also use the mean absolute deviation or MAD. This method uses the original units of the data, so interpretation will be easy. Calculating MAD is also very easy. You just need to follow these steps:
- Calculate the sample average
- Find the absolute deviation of each data point from the mean. You should ignore any negative signs.
- Find the average of all absolute deviations
While MAD has some benefits, the standard deviation is still the most commonly used measure of variability. One of its advantages is that it weights unevenly spread out samples more as compared to evenly spread out samples. That means you will be able to tell that the data is more unevenly spread out. Standard deviation also gives you a more precise measure of variability. It is also worth noting that standard deviation is more sensitive to outliers.
FAQs
Standard deviation is the average amount of variability in a dataset.
Low standard deviation means that the data points are clustered around the mean.
Yes, a high standard deviation shows that the data is less reliable as it is widely spread.
Variance is the degree of spread in a dataset. If there is more spread in the dataset, the variance will be large in relation to the mean.