Variation is also known as "Variability", "Dispersion", "Spread", and "Scatter". (5 names for one thing is one more example why statistics is confusing.) Variation is 1 of 3 major categories of measures describing a Distribution or data set. The others are Center (aka "Central Tendency") with measures like Mean, Mode, and Median and Shape (with measures like Skew and Kurtosis). Variation measures how "spread out" the data is.There are a number of different measures of Variation. This compare-and-contrast table shows the relative merits of each. - The Range is probably the least useful in statistics. It just tells you the highest and lowest values of a data set, and nothing about what's in between.
- The Interquartile Range (IQR) can be quite useful for visualizing the distribution of the data and for comparing several data sets -- as described in a
__recent post on this blog__. - Variance is the square of the Standard Deviation, and it is used as an interim step in the calculation of the latter. This squaring overly emphasizes the effects very high or very low values. Another drawback is that it is in units of the data squared (e.g. square kilograms, which can be meaningless). There is a Chi-Square Test for the Variance, and Variances are used in F tests and the calculations in ANOVA.
- The Mean Absolute Deviation is the average (unsquared) distance of the data points from the Mean. It is used when it is desirable to avoid emphasizing the effects of high and low values
- The Standard Deviation, being the square root of the Variance, does not overly emphasize the high and low values as the Variance does. Another major benefit is that it is in the same units as the data.
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Alpha is the Significance Level of a statistical test. We select a value for Alpha based on the level of Confidence we want that the test will avoid a False Positive (aka Alpha aka Type I) Error. In the diagrams below, Alpha is split in half and shown as shaded areas under the right and left tails of the Distribution curve. This is for a 2-tailed, aka 2-sided test. In the left graph above, we have selected the common value of 5% for Alpha. A Critical Value is the point on the horizontal axis where the shaded area ends. The Margin of Error (MOE) is half the distance between the two Critical Values.
A Critical Value is a value on the horizontal axis which forms the boundary of one of the shaded areas. And the Margin of Error is half the distance between the Critical Values. If we want to make Alpha even smaller, the distance between Critical Values would get even larger, resulting in a larger Margin of Error. The right diagram shows that if we want to make the MOE smaller, the price would be larger Alpha. This illustrates the Alpha - MOE see-saw effect. But what if we wanted a smaller MOE without making Alpha larger? Is that possible? It is -- by increasing n, the Sample Size. (It should be noted that, after a certain point, continuing to increase n yields diminishing returns. So, it's not a universal cure for these errors.)If you'd like to learn more about Alpha, I have 2 YouTube videos which may be of interest: |
## AuthorAndrew A. (Andy) Jawlik is the author of the book, Statistics from A to Z -- Confusing Concepts Clarified, published by Wiley. ## Archives
December 2020
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