By averaging squared deviations from the mean, variance can be calculated, which illustrates the degree of distribution in your data set.
The variance is more significant with respect to the mean when the data set has a higher degree of distribution.
Definition: Variance
To find the range of values in a data set for the average or mean, statisticians, employ the concept of mean square deviation. This may be by squaring the standard deviation.
For this, how stretched or squeezed a distribution is using this approach must be assessed. In statistics, sample and population discrepancies are the two types of variances that might exist.
Variance vs. standard deviation
The standard deviation is generated from the mean square deviation and indicates the average distance between each value and the mean. Specifically, it is the mean square deviation’s square root. Both metrics capture distributional variability, although they use different measurement units:
- While the mean square deviation is given in considerably greater quantities than the standard deviation (e.g., meters squared),
- The standard deviation is expressed in the same units as the original values (for example, meters).
It is more difficult to grasp the mean square deviation number intuitively since the mean square discrepancy’s units are substantially greater than those of a typical data set value. Because of this, the standard deviation is frequently chosen as the primary indicator of variability.
Conversely, the mean square deviation is utilized to draw statistical conclusions, since it provides more information on variability than the standard deviation.
Population vs. sample variance
In the following paragraphs, the difference between the population mean square deviation and sample variance is explained.
Population mean square deviation
You can obtain a precise estimate of the population mean square deviation once you have collected data from every member of the population in which you are interested.
It also reveals how evenly distributed data points are within a population by averaging the distances between each data point and the mean squared for that population.
Sample variance
The sample mean square deviation is used to estimate or draw conclusions about the population variation when data from a sample is collected. The amount of dispersion between the numbers in a list is measured by the sample mean square deviation ().
The mean square deviation will be minimal if all the numbers in a list are inside a small range of the expected values. The difference will be significantly greater if they are a long way apart. The sample mean square deviation is given by the equation:
Variance calculation: Step-by-step
Typically, the program you use for your statistical study will automatically calculate the mean square deviation. However, you may also perform a manual calculation to better comprehend how the formula functions.^{1}
When determining the mean square deviation manually, there are five key phases:
Step 1: Determine the mean
To find the mean, add up all the scores, then divide them by the number of scores.
Step 2: Find the score of the deviation from the mean
To determine the deviations from the mean, subtract the mean from each score.
Step 3: Square each deviation from the mean
Add up each deviation from the mean that produces a positive number.
Step 4: Sum up squares
The squared deviations are totaled and called the sum of squares.
Step 5: Divide the sum of squares by n – 1 or N
Divide the sum of the squares by (for a sample) or (for a population).
Reasons for variance
The mean square deviation is significant for two fundamental reasons:
- Mean square deviation is susceptible to parametric statistical tests.
- You can evaluate group differences by comparing a sample mean square deviations.
1. Homogeneity of variance in statistical tests
Prior to conducting parametric testing, variation must be considered. Also known as homogeneity of mean square deviation or homoscedasticity, these tests require identical or comparable variances when comparing various samples.
Test results are skewed and biased due to unequal variances between samples. Non-parametric tests are better suited if sample variances are uneven.
2. Using variance to assess group differences
The sample mean square deviation is used in statistical tests to evaluate group differences, such as variance tests and the analysis of variance (ANOVA). They evaluate whether the populations they represent are distinct from one another using the mean square deviations of the samples.
3. An ANOVA is used to evaluate group differences
The basic goal of an ANOVA is to evaluate variances within and across groups to determine whether group differences or individual differences can better account for the results.^{2}
The groups are probably different due to your treatment if the between-group mean square deviation is higher than the within-group mean square deviation. If not, the outcomes could originate from the sample members’ unique differences.
FAQs
The difference between the highest and lowest values is referred to as the range.
- Interquartile range: the range of a distribution’s middle half
- Standard deviation: the typical departure from the mean
- Mean square deviation: squared mean deviations are averaged out
The standard deviation is the average-squared deviation from the mean.
Both metrics capture distributional variability, although they use different measurement units. The units used to indicate standard deviation are the same as the values’ original ones, such as minutes or meters.
The sample discrepancy is used by statistical tests to evaluate population group differences, such as variance and the analysis of variance (ANOVA).
They determine whether the populations they represent significantly differ from one another using the sample variances.
Homoscedasticity, also known as homogeneity of the mean square deviation, is the presumption that variations in the groups being compared are equivalent or similar.
Because parametric statistical tests are sensitive to any differences, this is a crucial presumption. Results from tests are skewed and biased when the sample mean square deviation is uneven.
Sources
^{1} Wolter, Kirk M.. Introduction to mean square deviation estimation. Vol. 53. New York: Springer, 2007.
^{2} Bewick, Viv, Liz Cheek, and Jonathan Ball. “Statistics review 9: one-way analysis of variance.” Critical care 8, no. 2 (2004): 1-7.
^{3} Andersen, Torben G., Tim Bollerslev, and Ashish Das. “Variance‐ratio statistics and high‐frequency data: Testing for changes in intraday volatility patterns.” The Journal of Finance 56, no. 1 (2001): 305-327.