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How to find standard deviation?

Published in Statistics 3 mins read

Finding the standard deviation is a fundamental process in statistics that helps quantify the amount of variation or dispersion of a set of data values. It tells you, on average, how far each data point is from the mean of the dataset. A low standard deviation indicates that data points are generally close to the mean, while a high standard deviation suggests data points are spread out over a wider range of values.

Understanding Standard Deviation

Standard deviation is a crucial measure of data variability. It's widely used in various fields, from finance to quality control, to understand the consistency and predictability of data. Unlike simply looking at the range, standard deviation considers every data point, providing a more robust measure of spread.

For a deeper understanding of statistical terms, you can explore resources on basic statistical concepts.

Step-by-Step Guide to Calculating Standard Deviation

Calculating the standard deviation involves a series of straightforward steps that build upon each other. Here’s how to do it:

Step Action Explanation
1 Find the Mean Calculate the average of all data points in your dataset. Sum all values and divide by the total number of values.
2 Subtract the Mean Determine the deviation of each data point from the mean. Subtract the mean from each individual data point. Some results will be positive, others negative.
3 Square the Differences Take each of the differences calculated in Step 2 and square them. Squaring eliminates negative signs and gives more weight to larger deviations.
4 Calculate the Variance Find the average of these squared differences. This value is known as the variance. To get the variance, sum all the squared differences and divide by the total number of data points (for a population) or by one less than the total number of data points (for a sample).
5 Take the Square Root The final step is to calculate the square root of the variance obtained in Step 4. This result is your standard deviation.

Practical Example

Let's illustrate these steps with a simple dataset: [2, 4, 4, 4, 5]

Step 1: Find the Mean

  • Sum of data points = 2 + 4 + 4 + 4 + 5 = 19
  • Number of data points (n) = 5
  • Mean (average) = 19 / 5 = 3.8

Step 2: Subtract the Mean from Each Data Point

  • 2 - 3.8 = -1.8
  • 4 - 3.8 = 0.2
  • 4 - 3.8 = 0.2
  • 4 - 3.8 = 0.2
  • 5 - 3.8 = 1.2

Step 3: Square Each Difference

  • (-1.8)$^2$ = 3.24
  • (0.2)$^2$ = 0.04
  • (0.2)$^2$ = 0.04
  • (0.2)$^2$ = 0.04
  • (1.2)$^2$ = 1.44

Step 4: Calculate the Variance

  • Sum of squared differences = 3.24 + 0.04 + 0.04 + 0.04 + 1.44 = 4.8
  • Variance (assuming population, dividing by n) = 4.8 / 5 = 0.96

Step 5: Calculate the Standard Deviation

  • Standard Deviation = $\sqrt{0.96}$ $\approx$ 0.9798

This result tells us that, on average, the data points in the set [2, 4, 4, 4, 5] deviate by approximately 0.98 units from the mean of 3.8.