What is standard deviation & what does it represent in statistics? (with example)

© The Maths Studio (themathsstudio.net)
Sets of scores can have the same mean, but the scores themselves can be quite different.
For example, the set of scores 4,5,6 has a mean of 5, but so does the set of scores 3,5,7 and 1,5,9 and 0,5,10 and so on.
Clearly, the scores 4,5,6 are closer to the mean of 5 than the scores 1,5,9, which are much further away from the mean.
The standard deviation measures how far away, on average, each score is from the mean of those scores.
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Imagine you have a bunch of test scores from your classmates, and you want to understand how spread out or varied those scores are. Standard deviation is a number that helps you figure that out.
Think of it like this: If all the test scores are very similar, the standard deviation will be small. But if the scores are all over the place, the standard deviation will be larger.
So, standard deviation indicates how much the data points (like test scores) tend to differ from the average or mean. A small standard deviation means most of the data is close to the average, while a large standard deviation means the data is more spread out.
Technically speaking, standard deviation is a measure of how spread out or dispersed a set of data is. It quantifies the amount of variability or deviation from the mean (average) of the data.
Here's a more formal definition and explanation:
1. *Calculate the Mean (Average):* To find the standard deviation, start by calculating the mean of your data. The mean is the sum of all the values divided by the number of values.
2. *Find the Differences:* Next, find the difference between each data point and the mean. This tells you how far each point is from the average.
3. *Square the Differences:* To avoid negative differences canceling out positive differences, square each of the differences. This is an important step because it emphasizes how far each point is from the mean without regard to direction.
4. *Calculate the Variance:* Add up all these squared differences and divide by the number of data points. This gives you the variance, which is like the average of the squared differences.
5. *Take the Square Root:* Finally, to get the standard deviation, take the square root of the variance. The standard deviation measures the average "distance" of data points from the mean.
So, the standard deviation tells you how spread out or clustered your data is. A small standard deviation means the data points are close to the mean, while a large standard deviation means they are more spread out, indicating greater variability or dispersion. It's a valuable tool in statistics to understand the consistency or variability within a data set.
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