What Does High Variance Mean
- What Does A High Variance Mean
- How To Calculate Variance
- What Is A High Variance
- What Does High Variance Mean In Statistics
What Does Efficiency Variance Mean? Efficiency variance is the difference between the actual quantity of input put into a manufacturing process and the estimated or budgeted quantity. The input could be labor hours or other overhead costs. The efficiency variance shows how productive or efficient the manufacturing process was with its inputs. The variance (σ 2), is defined as the sum of the squared distances of each term in the distribution from the mean (μ), divided by the number of terms in the distribution (N). There's a more efficient way to calculate the standard deviation for a group of numbers, shown in the following equation. A high variance indicates that the data points are very spread out from the mean, and from one another. Variance is the average of the squared distances from each point to the mean. The process of finding the variance is very similar to finding the MAD, mean absolute deviation.
It would be useful to have a measure of scatter that has the following properties:
- The measure should be proportional to the scatter of the data (small when the data are clustered together, and large when the data are widely scattered).
- The measure should be independent of the number of values in the data set (otherwise, simply by taking more measurements the value would increase even if the scatter of the measurements was not increasing).
- The measure should be independent of the mean (since now we are only interested in the spread of the data, not its central tendency).
Both the variance and the standard deviation meet these three criteria for normally-distributed (symmetric, 'bell-curve') data sets.
The variance (σ2) is a measure of how far each value in the data set is from the mean. Here is how it is defined:
- Subtract the mean from each value in the data. This gives you a measure of the distance of each value from the mean.
- Square each of these distances (so that they are all positive values), and add all of the squares together.
- Divide the sum of the squares by the number of values in the data set.
The standard deviation (σ) is simply the (positive) square root of the variance.
The Summation Operator
In order to write the equation that defines the variance, it is simplest to use the summation operator, Σ. The summation operator is just a shorthand way to write, 'Take the sum of a set of numbers.' As an example, we'll show how we would use the summation operator to write the equation for calculating the mean value of data set 1. We'll start by assigning each number to variable, X1–X6, like this:
Think of the variable (X) as the measured quantity from your experiment—like number of leaves per plant—and think of the subscript as indicating the trial number (1–6). To calculate the average number of leaves per plant, we first have to add up the values from each of the six trials. Using the summation operator, we'd write it like this:
which is equivalent to:
or:
Obviously the sum is a lot more compact to write with the summation operator. Here is the equation for calculating the mean, μx, of our data set using the summation operator:
The general equation for calculating the mean, μ, of a set of numbers, X1 – XN, would be written like this:
Sometimes, for simplicity, the subscripts are left out, as we did on the right, above. Doing away with the subscripts makes the equations less cluttered, but it is still understood that you are adding up all the values of X.
The Equation Defining Variance
Now that you know how the summation operator works, you can understand the equation that defines the population variance (see note at the end of this page about the difference between population variance and sample variance, and which one you should use for your science project):
The variance (σ2), is defined as the sum of the squared distances of each term in the distribution from the mean (μ), divided by the number of terms in the distribution (N).
There's a more efficient way to calculate the standard deviation for a group of numbers, shown in the following equation:
You take the sum of the squares of the terms in the distribution, and divide by the number of terms in the distribution (N). From this, you subtract the square of the mean (μ2). It's a lot less work to calculate the standard deviation this way.
It's easy to prove to yourself that the two equations are equivalent. Start with the definition for the variance (Equation 1, below). Expand the expression for squaring the distance of a term from the mean (Equation 2, below).
Now separate the individual terms of the equation (the summation operator distributes over the terms in parentheses, see Equation 3, above). In the final term, the sum of μ2/N, taken N times, is just Nμ2/N.
Next, we can simplify the second and third terms in Equation 3. In the second term, you can see that ΣX/N is just another way of writing μ, the average of the terms. So the second term simplifies to −2μ2 (compare Equations 3 and 4, above). In the third term, N/N is equal to 1, so the third term simplifies to μ2 (compare Equations 3 and 4, above).
Finally, from Equation 4, you can see that the second and third terms can be combined, giving us the result we were trying to prove in Equation 5.
As an example, let's go back to the two distributions we started our discussion with:
data set 2: 1, 2, 4, 5, 7, 11 .
What are the variance and standard deviation of each data set?
We'll construct a table to calculate the values. You can use a similar table to find the variance and standard deviation for results from your experiments.
| Data Set | N | ΣX | ΣX2 | μ | μ2 | σ2 | σ |
|---|---|---|---|---|---|---|---|
| 1 | 6 | 30 | 166 | 5 | 25 | 2.67 | 1.63 |
| 2 | 6 | 30 | 216 | 5 | 25 | 11.00 | 3.32 |
Although both data sets have the same mean (μ = 5), the variance (σ2) of the second data set, 11.00, is a little more than four times the variance of the first data set, 2.67. The standard deviation (σ) is the square root of the variance, so the standard deviation of the second data set, 3.32, is just over two times the standard deviation of the first data set, 1.63.
A histogram showing the number of plants that have a certain number of leaves. All plants have a different number of leaves ranging from 3 to 8 (except for 2 plants that have 4 leaves). The difference between the highest number of leaves and lowest number of leaves is 5 so the data has relative low variance.
A histogram showing the number of plants that have a certain number of leaves. All plants have different number of leaves ranging from 1 to 11. The difference between the plant with the highest number of leaves and the lowest number of leaves is 10, so the data has relatively high variance.
The variance and the standard deviation give us a numerical measure of the scatter of a data set. These measures are useful for making comparisons between data sets that go beyond simple visual impressions.
Population Variance vs. Sample Variance
The equations given above show you how to calculate variance for an entire population. However, when doing science project, you will almost never have access to data for an entire population. For example, you might be able to measure the height of everyone in your classroom, but you cannot measure the height of everyone on Earth. If you are launching a ping-pong ball with a catapult and measuring the distance it travels, in theory you could launch the ball infinitely many times. In either case, your data is only a sample of the entire population. This means you must use a slightly different formula to calculate variance, with an N-1 term in the denominator instead of N:
This is known as Bessel's correction.
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Deviation just means how far from the normal
Standard Deviation
The Standard Deviation is a measure of how spread out numbers are.
Its symbol is σ (the greek letter sigma)
The formula is easy: it is the square root of the Variance. So now you ask, 'What is the Variance?'
Variance
The Variance is defined as:
The average of the squared differences from the Mean.
To calculate the variance follow these steps:
- Work out the Mean (the simple average of the numbers)
- Then for each number: subtract the Mean and square the result (the squared difference).
- Then work out the average of those squared differences. (Why Square?)
Example
You and your friends have just measured the heights of your dogs (in millimeters):
The heights (at the shoulders) are: 600mm, 470mm, 170mm, 430mm and 300mm.
Find out the Mean, the Variance, and the Standard Deviation.
Your first step is to find the Mean:
Answer:
| Mean | = | 600 + 470 + 170 + 430 + 3005 |
| = | 19705 | |
| = | 394 |
so the mean (average) height is 394 mm. Let's plot this on the chart:
Now we calculate each dog's difference from the Mean:
To calculate the Variance, take each difference, square it, and then average the result:
| Variance | ||
| σ2 | = | 2062 + 762 + (−224)2 + 362 + (−94)25 |
| = | 42436 + 5776 + 50176 + 1296 + 88365 | |
| = | 1085205 | |
| = | 21704 | |
So the Variance is 21,704
And the Standard Deviation is just the square root of Variance, so:
| Standard Deviation | ||
| σ | = | √21704 |
| = | 147.32... | |
| = | 147(to the nearest mm) | |
And the good thing about the Standard Deviation is that it is useful. Now we can show which heights are within one Standard Deviation (147mm) of the Mean:
So, using the Standard Deviation we have a 'standard' way of knowing what is normal, and what is extra large or extra small.
Rottweilers are tall dogs. And Dachshunds are a bit short, right?
Using
We can expect about 68% of values to be within plus-or-minus1 standard deviation.
Read Standard Normal Distribution to learn more.
Also try the Standard Deviation Calculator.
But ... there is a small change with Sample Data
Our example has been for a Population (the 5 dogs are the only dogs we are interested in).
But if the data is a Sample (a selection taken from a bigger Population), then the calculation changes!
When you have 'N' data values that are:
- The Population: divide by N when calculating Variance (like we did)
- A Sample: divide by N-1 when calculating Variance
All other calculations stay the same, including how we calculated the mean.
Example: if our 5 dogs are just a sample of a bigger population of dogs, we divide by 4 instead of 5 like this:
Think of it as a 'correction' when your data is only a sample.
Formulas
Here are the two formulas, explained at Standard Deviation Formulas if you want to know more:
What Does A High Variance Mean
|
| The 'Sample Standard Deviation': |
Looks complicated, but the important change is to
divide by N-1 (instead of N) when calculating a Sample Variance.
*Footnote: Why square the differences?
If we just add up the differences from the mean ... the negatives cancel the positives:
| 4 + 4 − 4 − 44 = 0 |
So that won't work. How about we use absolute values?
| 4 + 4 + −4 + −4 4 = 4 + 4 + 4 + 44 = 4 |
That looks good (and is the Mean Deviation), but what about this case:
How To Calculate Variance
| 7 + 1 + −6 + −2 4 = 7 + 1 + 6 + 24 = 4 |
Oh No! It also gives a value of 4, Even though the differences are more spread out.
What Is A High Variance
So let us try squaring each difference (and taking the square root at the end):
| √(42 + 42 + (-4)2 + (-4)24) = √(644) = 4 |
| √(72 + 12 + (-6)2 + (-2)24) = √(904) = 4.74... |
That is nice! The Standard Deviation is bigger when the differences are more spread out ... just what we want.
In fact this method is a similar idea to distance between points, just applied in a different way.
What Does High Variance Mean In Statistics
And it is easier to use algebra on squares and square roots than absolute values, which makes the standard deviation easy to use in other areas of mathematics.