# standard deviation

noun

## Definition of standard deviation

1 : a measure of the dispersion of a frequency distribution that is the square root of the arithmetic mean of the squares of the deviation of each of the class frequencies from the arithmetic mean of the frequency distribution also : a similar quantity found by dividing by one less than the number of squares in the sum of squares instead of taking the arithmetic mean
2 : a parameter that indicates the way in which a probability function or a probability density function is centered around its mean and that is equal to the square root of the moment in which the deviation from the mean is squared

## Examples of standard deviation in a Sentence

Recent Examples on the Web Nuvrei’s croissants are a standard deviation away from Portland’s croissant norm. Notre Dame, which lost 100 students since the last two-year academic placements, could remain playing football in 5A. Hines said the AIA looks at what conferences the school are playing at now to conduct the standard deviation. As expected from past studies, the students in the active learning classes consistently outperformed their peers (and themselves), scoring a half a standard deviation higher on the quizzes. According to Live Science, the team reached its conclusion with a standard deviation of 3.5 sigma, far below the 5 sigma confidence interval that physicists rely on in order to confirm a theory. Using standard deviation, the teams will be put into conferences. Overall, boys performed marginally better than girls in math, by about 10% of a standard deviation—suggesting only a slight gender gap. As a result, the mean and the standard deviation for each characteristic should be about the same—but not too identical. Using Twitter reduced performance on the test by about 25 to 40 percent of a standard deviation from the average result, as the paper explains.

These example sentences are selected automatically from various online news sources to reflect current usage of the word 'standard deviation.' Views expressed in the examples do not represent the opinion of Merriam-Webster or its editors. Send us feedback.

## First Known Use of standard deviation

1894, in the meaning defined at sense 1

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Time Traveler for standard deviation ## The first known use of standard deviation was in 1894

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Standard English

## Statistics for standard deviation

Last Updated

1 Dec 2019

Cite this Entry

“Standard deviation.” The Merriam-Webster.com Dictionary, Merriam-Webster Inc., https://www.merriam-webster.com/dictionary/standard%20deviation. Accessed 13 December 2019.

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More Definitions for standard deviation

standard deviation

noun

## Financial Definition of standard deviation

What It Is

Standard deviation is a measure of how much an investment's returns can vary from its average return. It is a measure of volatility and in turn, risk. The formula for standard deviation is:

Standard Deviation = [1/n *  (ri - rave)2]½

where:
ri = actual rate of return
rave = average rate of return
n = number of time periods

For math-oriented readers, standard deviation is the square root of the variance.

How It Works

Let's assume that you invest in Company XYZ stock, which has returned an average 10% per year for the last 10 years. How risky is this stock compared to, say, Company ABC stock? To answer this, let's first take a closer look at the year-by-year returns that compose that average: At first look, we can see that the average return for both stocks over the last 10 years was indeed 10%. But let's look in a different way at how close XYZ's returns in any given year were to the average 10%: As you can see, only during year 9 did XYZ return the average 10%. In the other years, the return was higher or lower -- sometimes much higher (as in year 7) or much lower (as in year 2). Now look at the annual returns on Company ABC stock, which also had a 10% average return for the last 10 years: As you can see, Company ABC also averaged 10% return over 10 years but did so with far less variance than Company XYZ. Its returns are more tightly clustered around that 10% average. Thus, we can say that Company XYZ is more volatile than Company ABC stock. Standard deviation seeks to measure this volatility by calculating how "far away" the returns tend to be from the average over time.

For instance, let's calculate the standard deviation for Company XYZ stock. Using the formula above, we first subtract each year's actual return from the average return, then square those differences (that is, multiply each difference by itself): Next, we add up column D (the total is 3,850). We divide that number by the number of time periods minus one (10-1=9; this is called the "nonbiased" approach and it is important to remember that some calculate standard deviation using all time periods -- 10 in this case rather than 9). Then we take the square root of the result. It looks like this:

Standard deviation = √(3,850/9) = √427.78 = 0.2068

Using the same process, we can calculate that the standard deviation for the less volatile Company ABC stock is a much lower 0.0129.

Why It Matters

Standard deviation is a measure of risk that an investment will not meet the expected return in a given period. The smaller an investment's standard deviation, the less volatile (and hence risky) it is. The larger the standard deviation, the more dispersed those returns are and thus the riskier the investment is.

Many technical indicators, such as Bollinger Bands, incorporate the notion of standard deviation as a way to determine whether to buy or sell a stock, but it is important to remember that standard deviation is only one of many measures of risk and should not be the last word in deciding whether a stock is "too risky" or "not risky enough."

More from Merriam-Webster on standard deviation

Britannica.com: Encyclopedia article about standard deviation

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