Variance definition, the state, quality, or fact of being variable, divergent, different, or anomalous. Preserving the total variance is not the only (or even an important) reason to require that the change of variables is an orthogonal transformation. As we saw earlier, the transformation (c cdot I ) inflates the total variance by the factor of (c^2 ), but it does so by uniformly inflating the variance of each variable. So when we compute the. 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. Variance is a measure of how much values in a data set differ from the mean. Do scores tend to center around the mean or are they spread out? For example, take two data sets each with 7 scores ranging from 1 to 9 and a mean of 5. The set with scores 1, 1, 3, 5, 7, 9, 9 has greater variance than the set with scores 1, 3, 5, 5, 5, 7, 9.
- What Does A High Variance Mean
- Sample Variance
- Define Variance In Statistics
- What Does High Variance Mean
- What Does High Variance Mean In Statistics
Learning Objectives
- State the difference in bias between (η^2) and (ω^2)
- Compute (η^2) Compute (ω^2)
- Distinguish between (ω^2) and partial (ω^2)
- State the bias in (R^2) and what can be done to reduce it
![What does high variance mean in statistics What does high variance mean in statistics](https://www.riskprep.com/images/stories/miscimages/kurt4.png)
Effect sizes are often measured in terms of the proportion of variance explained by a variable. In this section, we discuss this way to measure effect size in both ANOVA designs and in correlational studies.
ANOVA Designs
Responses of subjects will vary in just about every experiment. Consider, for example, the 'Smiles and Leniency' case study. A histogram of the dependent variable 'leniency' is shown in Figure (PageIndex{1}). It is clear that the leniency scores vary considerably. There are many reasons why the scores differ. One, of course, is that subjects were assigned to four different smile conditions and the condition they were in may have affected their leniency score. In addition, it is likely that some subjects are generally more lenient than others, thus contributing to the differences among scores. There are many other possible sources of differences in leniency ratings including, perhaps, that some subjects were in better moods than other subjects and/or that some subjects reacted more negatively than others to the looks or mannerisms of the stimulus person. You can imagine that there are innumerable other reasons why the scores of the subjects could differ.
One way to measure the effect of conditions is to determine the proportion of the variance among subjects' scores that is attributable to conditions. In this example, the variance of scores is (2.794). The question is how this variance compares with what the variance would have been if every subject had been in the same treatment condition. We estimate this by computing the variance within each of the treatment conditions and taking the mean of these variances. For this example, the mean of the variances is (2.649). Since the mean variance within the smile conditions is not that much less than the variance ignoring conditions, it is clear that 'Smile Condition' is not responsible for a high percentage of the variance of the scores. The most convenient way to compute the proportion explained is in terms of the sum of squares 'conditions' and the sum of squares total. The computations for these sums of squares are shown in the chapter on ANOVA. For the present data, the sum of squares for 'Smile Condition' is (27.535) and the sum of squares total is (377.189). Therefore, the proportion explained by 'Smile Condition' is:
What Does A High Variance Mean
[frac{27.535}{377.189} = 0.073]
Thus, (0.073) or (7.3%) of the variance is explained by 'Smile Condition.'
An alternative way to look at the variance explained is as the proportion reduction in error. The sum of squares total ((377.189)) represents the variation when 'Smile Condition' is ignored and the sum of squares error ((377.189 - 27.535 = 349.654)) is the variation left over when 'Smile Condition' is accounted for. The difference between (377.189) and (349.654) is (27.535). This reduction in error of (27.535) represents a proportional reduction of (27.535/377.189 = 0.073), the same value as computed in terms of proportion of variance explained.
This measure of effect size, whether computed in terms of variance explained or in terms of percent reduction in error, is called (η^2) where (η) is the Greek letter eta. Unfortunately, (η^2) tends to overestimate the variance explained and is therefore a biased estimate of the proportion of variance explained. As such, it is not recommended (despite the fact that it is reported by a leading statistics package).
An alternative measure, (ω^2) (omega squared), is unbiased and can be computed from
[omega ^2 = frac{SSQ_{condition}-(k-1)MSE}{SSQ_{total}+MSE}]
where (MSE) is the mean square error and (k) is the number of conditions. For this example, (k = 4) and (ω^2 = 0.052).
It is important to be aware that both the variability of the population sampled and the specific levels of the independent variable are important determinants of the proportion of variance explained. Consider two possible designs of an experiment investigating the effect of alcohol consumption on driving ability. As can be seen in Table (PageIndex{1}), (text{Design 1}) has a smaller range of doses and a more diverse population than (text{Design 2}). What are the implications for the proportion of variance explained by Dose? Variation due to Dose would be greater in (text{Design 2}) than (text{Design 1}) since alcohol is manipulated more strongly than in (text{Design 1}). However, the variance in the population should be greater in (text{Design 1}) since it includes a more diverse set of drivers. Since with (text{Design 1}) the variance due to Dose would be smaller and the total variance would be larger, the proportion of variance explained by Dose would be much less using (text{Design 1}) than using (text{Design 2}). Thus, the proportion of variance explained is not a general characteristic of the independent variable. Instead, it is dependent on the specific levels of the independent variable used in the experiment and the variability of the population sampled.
Design | Dose | Casinos in us. Population |
1 | 0.00 | All Drivers between 16 and 80 Years of Age |
0.30 | ||
0.60 | ||
2 | 0.00 | Experienced Drivers between 25 and 30 Years of Age |
0.50 | ||
1.00 |
Factorial Designs
In one-factor designs, the sum of squares total is the sum of squares condition plus the sum of squares error. The proportion of variance explained is defined relative to sum of squares total. In an (A times B) design, there are three sources of variation ((A, B, A times B)) in addition to error. The proportion of variance explained for a variable ((A), for example) could be defined relative to the sum of squares total ((SSQ_A + SSQ_B + SSQ_{Atimes B} + SSQ_{error})) or relative to (SSQ_A + SSQ_{error}).
To illustrate with an example, consider a hypothetical experiment on the effects of age ((6) and (12) years) and of methods for teaching reading (experimental and control conditions). The means are shown in Table (PageIndex{2}). The standard deviation of each of the four cells ((Age times Treatment) combinations) is (5). (Naturally, for real data, the standard deviations would not be exactly equal and the means would not be whole numbers.) Finally, there were (10) subjects per cell resulting in a total of (40) subjects.
Treatment | ||
---|---|---|
Age | Experimental | Control |
6 | 40 | 42 |
12 | 50 | 56 |
The sources of variation, degrees of freedom, and sums of squares from the analysis of variance summary table as well as four measures of effect size are shown in Table (PageIndex{3}). Note that the sum of squares for age is very large relative to the other two effects. This is what would be expected since the difference in reading ability between (6)- and (12)-year-olds is very large relative to the effect of condition.
Source | df | SSQ | (η^2) | partial (η^2) | (ω^2) | partial (ω^2) |
---|---|---|---|---|---|---|
Age | 1 | 1440 | 0.567 | 0.615 | 0.552 | 0.586 |
Condition | 1 | 160 | 0.063 | 0.151 | 0.053 | 0.119 |
A x C | 1 | 40 | 0.016 | 0.043 | 0.006 | 0.015 |
Error | 36 | 900 | ||||
Total | 39 | 2540 |
First, we consider the two methods of computing (η^2), labeled (η^2) and partial (η^2). Gals sports betting on line. The value of (η^2) for an effect is simply the sum of squares for this effect divided by the sum of squares total. For example, the (η^2) for Age is (1440/2540 = 0.567). As in a one-factor design, (η^2) is the proportion of the total variation explained by a variable. Partial (η^2) for Age is (SSQ_{Age}) divided by ((SSQ_{Age} + SSQ_{error})), which is (1440/2340 = 0.615).
As you can see, the partial (η^2) is larger than (η^2). This is because the denominator is smaller for the partial (η^2). The difference between (η^2) and partial (η^2) is even larger for the effect of condition. This is because (SSQ_{Age}) is large and it makes a big difference whether or not it is included in the denominator.
As noted previously, it is better to use (ω^2) than (η^2) because (η^2) has a positive bias. You can see that the values for (ω^2) are smaller than for (η^2). The calculations for (ω^2) are shown below:
[omega ^2 = frac{SSQ_{effect}-df_{effect}MS_{error}}{SSQ_{total}+MS_{error}}]
[omega _{partial}^2 = frac{SSQ_{effect}-df_{effect}MS_{error}}{SSQ_{effect}+(N-df_{effect})MS_{error}}]
where (N) is the total number of observations.
The choice of whether to use (ω^2) or the partial (ω^2) is subjective; neither one is correct or incorrect. However, it is important to understand the difference and, if you are using computer software, to know which version is being computed. (Beware, at least one software package labels the statistics incorrectly).
Correlational Studies
In the section 'Partitioning the Sums of Squares' in the Regression chapter, we saw that the sum of squares for (Y) (the criterion variable) can be partitioned into the sum of squares explained and the sum of squares error. The proportion of variance explained in multiple regression is therefore:
Sample Variance
[SSQ_{explained}/SSQ_{total }]
Define Variance In Statistics
In simple regression, the proportion of variance explained is equal to (r^2); in multiple regression, it is equal to (R^2).
In general, (R^2) is analogous to (η^2) and is a biased estimate of the variance explained. The following formula for adjusted (R^2) is analogous to (ω^2) and is less biased (although not completely unbiased):
[R_{adjusted}^{2} = 1 - frac{(1-R^2)(N-1)}{N-p-1}]
where (N) is the total number of observations and (p) is the number of predictor variables.
Contributor
- Online Statistics Education: A Multimedia Course of Study (http://onlinestatbook.com/). Project Leader: David M. Lane, Rice University.
What Does High Variance Mean
What does a high/low standard deviation mean in real terms?
1 Answer
The higher the standard deviation the more variability or spread you have in your data.
Explanation:
Standard deviation measures how much your entire data set differs from the mean.
The larger your standard deviation, the more spread or variation in your data. Small standard deviations mean that most of your data is clustered around the mean.
In the following graph, the mean is 84.47, the standard deviation is 6.92 and the distribution looks like this:
Many of the test scores are around the average. There's one student who scored a 96, two students who scored 69, another two who scored 71, but most students scored close to somewhat close to the average of 84.47.
Many of the test scores are around the average. There's one student who scored a 96, two students who scored 69, another two who scored 71, but most students scored close to somewhat close to the average of 84.47.
In this second graph, the mean is 80, the standard deviation is 14.57 , and the distribution looks like this:
There is greater variability in the test scores. One student scored a 24, which is pretty far from the average test score of 80. Another student scored a 45, which also isn't close to 80.
There is greater variability in the test scores. One student scored a 24, which is pretty far from the average test score of 80. Another student scored a 45, which also isn't close to 80.
Related topic
Mean and Standard Deviation of a Probability Distribution
Questions
Questions