Which measure of variability is a measure of the average distance from the mean?

Two of the most popular ways to measure variability or volatility in a set of data are standard deviation and average deviation, also known as mean absolute deviation. Though the two measurements are similar, they are calculated differently and offer slightly different views of data.

Determining volatility—that is, deviation from the center—is important in finance, so professionals in accounting, investing, and economics should be familiar with both concepts.

Key Takeaways

• Standard deviation is the most common measure of variability and is frequently used to determine the volatility of financial instruments and investment returns.
• Standard deviation is considered the most appropriate measure of variability when using a population sample, when the mean is the best measure of center, and when the distribution of data is normal.
• Some argue that average deviation, or mean absolute deviation, is a better gauge of variability when there are distant outliers or the data is not well distributed.

Understanding Standard Deviation

Standard deviation is the most common measure of variability and is frequently used to determine the volatility of markets, financial instruments, and investment returns. To calculate the standard deviation:

1. Find the mean, or average, of the data points by adding them and dividing the total by the number of data points.
2. Subtract the mean from each data point and square the difference of each result.
3. Find the mean of those squared differences and then the square root of the mean.

Squaring the differences between each point and the mean avoids the issue of negative differences for values below the mean, but it means the variance is no longer in the same unit of measure as the original data. Taking the square root means the standard deviation returns to the original unit of measure and is easier to interpret and use in further calculations.

Average Deviation

The average deviation, or mean absolute deviation, is calculated similarly to standard deviation, but it uses absolute values instead of squares to circumvent the issue of negative differences between the data points and their means.

To calculate the average deviation:

1. Calculate the mean of all data points.
2. Calculate the difference between the mean and each data point.
3. Calculate the average of the absolute values of those differences.

Standard Deviation Versus Average Deviation

Standard deviation is often used to measure the volatility of returns from investment funds or strategies because it can help measure volatility. Higher volatility is generally associated with a higher risk of losses, so investors want to see higher returns from funds that generate higher volatility. For example, a stock index fund should have relatively low standard deviation compared with a growth fund.

The mean average, or mean absolute deviation, is considered the closest alternative to standard deviation. It is also used to gauge volatility in markets and financial instruments, but it is used less frequently than standard deviation.

According to mathematicians, when a data set is of normal distribution—that is, there aren't many outliers—standard deviation is generally the preferable gauge of variability. But when there are large outliers, standard deviation registers higher levels of dispersion (or deviation from the center) than mean absolute deviation.

So, what variability refers to is how dispersed or spread out the data values are, or looking at it from another point of view how wide the data distribution is when it is graphed. If all data values are the same, then, of course, there is zero variability. The graph of the distribution would have zero width. If all the values lie very close to each other there is little variability and the distribution's graph would be quite narrow. If, on the other hand, the numbers are spread out all over the place, there is more variability and the graph would be wider.

VariabilityPractice
Exercise 2:Variability has to do with the:

No Response
Height of a distribution
Width of a distribution
Curvature of a distribution
Lopsidedness of a distribution

Lesson 1: Summary Measures of Data 1.4 - 3

---

Biostatistics for the Clinician

Again, it turns out as was the case with measures of central tendency, that there are many measures of variability. In the medical research literature some of the most frequently used measures are the standard deviation, interquartile range, and the range (see Figure 2.5).VariabilityPractice
Exercise 3:Variability is measured by:

No Response
The standard deviation
The interquartile range
The range
All except "No Response" above

Lesson 1: Summary Measures of Data 1.4 - 4

---

Biostatistics for the Clinician

One way to measure the spread of information or data is by looking at the standard deviation. It's just the mean spread which you extract from the information (see the standard deviation formula below).Standard Deviation Formula

To get the standard deviation, as you can see in the formula, first you square the distances values are from the mean. Then you sum those squared differences. Then you divide that sum by the number of differences. Finally, you take the square root of that quotient. The reaon that you subtract and square is pretty clear. Whether the value is above the mean or below the mean the squared difference between the value and the mean comes out the same when it is squared. So positive and negative makes no difference here. If you didn't square, they would tend to cancel each other out. When you divide by the number of values to get an average you find the square root of the whole thing because, it was squared earlier, to get back to the original measures. In other words by squaring to get rid of the negative and positive values you get squared measures. So you take the square root to get back to the original more intuitive kinds of measures like feet, cubic inches, or whatever else it might be. The standard deviation can be thought of as the average distance that values are from the mean of the distribution (see the standard deviation formula above).

VariabilityPractice
Exercise 4:The standard deviation measures:

No Response
Average distance from the mean
Distance between top and bottom values
Distance between 1st and 3rd quartiles
None of the above

Of course, given the formula, to compute a standard deviation you must be able to compute a meaningful mean. Consequently, computation of the standard deviation requires interval or ratio variables. Furthermore, in a distribution having a bell (normal) curve, it always turns out that when you know the standard deviation, you also know that approximately 68% of the values lie within 1 standard deviation of the mean. You also know that approximately 2.1% of the values lie in each tail of the distribution beyond 2 standard deviations from the mean (again see Figure 2.5).

VariabilityPractice
Exercise 5:In a normal distribution the percentage of scores within 1 standard deviation of the mean is approximately:

No Response
2.1%
5%
68%
95.8%

Lesson 1: Summary Measures of Data 1.4 - 5

---

Biostatistics for the Clinician

You should recall that the median is the point in the distribution that 50% of the sample is below and 50% is above. In other words the median is at the 50th percentile. Quartiles can also be defined. The 1st quartile is at the 25th percentile. The 2nd quartile is at the 50th percentile. The 3rd quartile is at the 75th percentile. And, the 4th quartile is at the 100th percentile.

The interquartile range then extends from the 25th percentile to the 75th percentile. It includes 50% of the values in the sample. So, the interquartile range is the distance between the 25th percentile and the 75th percentile. The interquartile range then is another measure of variability. But unlike the standard deviation, it can be appropriately applied with ordinal variables. Therefore it is used especially in conjunction with nonparametric statistics (see the interquartile range in the figure below).

Ranges

So, another way to display data that's been proposed by exploratory data analysis is to rank the data from low to high, then find the median and then the quartile values, the values between which one half of the data resides. When you do this you can then plot a box plot containing half the data (see the figure below). The rest of the data is out in the wings. And, you can see the interquartile range which contains those values between the lower and upper quartiles. You'll see more explicit clinical medicine examples of this in Lesson 1.5.

Which measures of variability is the average of squared distances from the mean?

Which measure of variability describes the typical distance of scores from the mean?

Which measure of dispersion is a measure of the average distance from the mean?

What is median variability?