Tag Archives: Standard Deviation


Normal Curve (Product Adoption)

When a new and good product hits the market, how fast or slow are you to buy it? Some people get it immediately. Others wait for varying lengths of time before making their purchase decision.

According to one Internet website (www.quickmba.com), consumers can be classified into 5 categories based on how quickly they acquire new items. A picture of the famous bell-shaped curve, like the one shown here, indicated the descriptive labels and sizes of the 5 groups.

By considering the percentage of people in each of the 5 groups (as well as the position of the short, dark “notches” on the bell curve’s baseline), you should be able to discern that the statistical concepts of mean and standard deviation were used to “define” each group. For example, a person would be classified as an Early Adopter if he/she tends to purchase new products with a speed that’s between 1 and 2 SDs faster than average.

It is interesting to note that there are 3 sections on the left side of this bell curve but only 2 on the right. The pink area begins 1 SD from the mean and extends all the way to the right. Thus, the percentage of Laggards is equal to the combined percentages of Innovators and Early Adopters. Some people, if creating this picture anew, might split the pink area into 2 parts (thus forming a total of 6 sections rather than 5), with the percentage of Laggards equal to the percentage of Innovators.

To see the original discussion of what was called the “Product Diffusion Curve,” go to http://www.quickmba.com/marketing/product/diffusion/

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Mystery Person

This important historical figure:

a) … coined the term “standard deviation”;

b) … had his book, “The Grammar of Science,” selected by 23-year-old Albert Einstein as the first item to be studied in Einstein’s elite “book club”;

c) … co-founded (1885) the “Men & Women’s Club” to discuss gender equality;

d) … has his main stats creation summarized by his last name’s middle letter.

FOR THE ANSWER: go to www.rutherfordjournal.org/article010107.html

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The standard deviation is almost always smaller than the variance, as the former is equal to the square root of the latter. For example, if the variance is 25, SD = 5. But consider what happens if the variance is between 0 and 1. In these fully legitimate situations, SD > variance.

Do real data ever produce variances and SDs that are smaller than 1.00? Most certainly! Here are 5 examples:

  •  Statements in an attitude inventory typically are set up in a Likert format, with response-options extending from “strongly agree” to “strongly disagree” and scored 1 through 5. The variance and SD of the responses to any given item usually are < 1.00.
  • The correlation between 2 variables is often computed separately for subgroups, with the mean r reported along with the SD or variance of the correlation coefficients. This SD or variance must be < 1.00.
  • Both the SD and variance of a set of proportions necessarily will turn out to be < 1.00
  • In test-development efforts, the mean item discrimination for subgroups of items (or for all items) is often reported, along with the SD or variance of these item indices. Because item discrimination can range from 0-to-1, the variability of a set of these values must be < 1.00.
  • In meta-analysis investigations that combine the results of different factor analytic studies, the mean factor loading is sometimes reported for each factor along with the SD or variance (that must be < 1.00) of the given factor’s loadings.

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Many people think a standard deviation indicates the “standard” amount that individual numbers deviate from the group’s mean. Specifically, they think an SD is computed as the average (i.e., arithmetic mean) of the deviation scores, disregarding whether the original scores are above or below the mean. Not so. For most groups of numbers, the SD is about 1.25 times as large as the “average deviation from the mean.”

Consider, for example, this population of 10 scores: 1, 2, 3, 4, 5, 5, 6, 7, 8, and 9. Disregarding sign, the average deviation from the mean = 2.00. However, the SD = approximately 2.45. The SD is larger because it gives greater weight to scores that lie farther away from the mean. It does this by squaring the deviations. The SD is computed as the “root-mean-squared-deviation,” with these 4 words explaining, in reverse order, what you must do to calculate the SD: (1) figure out how far each original score deviates from the mean, (2) square each of these deviation scores, (3) take the mean of the squared deviations, (4) compute the square root of the result arrived at in Step 3.

For more information about the standard deviation, go to http://en.wikipedia.org/wiki/Standard_deviation.

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