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Measures of Dispersion Explained

measures of dispersion -magoosh

How normal is your data? I mean, does your sample seem to represent the population well? Does your data have a regular pattern? Is your data precise or vague? In statistics, we answer these questions using measures of central tendency and measures of dispersion. There are three big measures that help determine if your data is normal and precise. Let’s explore them.

Range

The range of your data gives you good insight into all the measurements that are covered. Unlike the median, which reveals the middle value, the range gives you an idea about the size of your measurements.

The range is very simple to calculate. It is merely the greatest measurement minus the lowest measurement. It allows you to see the numeric distance covered by the data.

When compared to the mean, median, and mode, the range also lets you identify outliers. Outliers are values that are very high or low and far from the mean, which is the general model of the data.

Interquartile Range

The interquatile range gives you a great picture of the data. Now, as the name suggests, the data is divided into four sections: Q1, Q2, Q3, and Q4.

Q1 represents the range from the lowest value to the median value of the first half of the data. Q2 is the range from the median of the first half of the data to the median of the entire data set. Q3 is the range from the median of the data set to the median value of the second half of the data set. Q4 is the range from the median of the second half of the data set to the greatest value. The interquartile range is Q3 – Q2.

The sizes of the interquartile ranges give you insight into the variability of the data set. Ideally, they would all be equal or close to equal. If they vary a lot, then your data may be skewed. As a measure of dispersion, the range gives a lot of information about the data.

Standard Deviation

Perhaps one of the most widely used measures of dispersion is standard deviation. Standard deviation is a great way to get a sense of the variability of the data. It is a measure of the proportions of the data set. It is represented by s for a sample, or σ for a population.

In a very basic sense, the standard deviation gives you sense of how the actual values of the data set vary from the mean. A high standard deviation means that the data set vary a lot, but a low standard deviation means that the data do not vary very much. the smaller the standard deviation, the better.

The standard deviation of a sample is calculated by

Since you can never truly know the standard deviation of a population, you can only estimate it.

Variance

Variance is similar to standard deviation. In fact, you can easily calculate one from another. Essentially, variance is a more precise measure of how precise your data is. It is represented by s2 for a sample and σ2 for a population.

If you know the standard deviation of the data, then the variance is easily calculated as the square of the standard deviation, for both the sample and the population. If you do not know the standard deviation, then the formal formula is

The variance is a very precise measure of how spread out the data is. For example, let’s say that you have a set of twelve measurements that are all the same. In this case, the variance would be zero since the values are all really close together (the same in fact). But if you have a set of data like A = {1,1,2,3,5,6,8,8,9,9,11,12} then the variance is 14.75. This means that the data is very spread out, especially since it is higher than the highest value.

In general, you want a variance that is close to zero. This means that the data is very precise.

The Takeaways

When it comes to measures of dispersion, there are three concepts that give you an idea of how dispersed your data is. The range gives an overall picture of how widely spread the data is. Standard deviation gives an idea of how close together the data is compared to the mean. Variance is the most precise measure of how dispersed your data really is.

I hope that this post helped to clarify measures of dispersion. I also can’t wait to see what kinds of questions you have below. Happy statistics!

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