Chapter 8:Shapes of Distributions

The Normal Distribution

(often referred to as a "bell-shaped" curve) is a distribution that often occurs in naturally occurring phenomena. Heights and weights of humans are generally normally distributed. The yields of corn are often normal. The diameters of apples are generally normally distributed. Pay is usually not normally distributed, nor is performance.

Other Examples of Shapes of Distributions - Performance Ratings Pg. 8.7

-Distributions can take on different shapes, and we can better understand the characteristics of the data based on their shapes. -Notice that marketing is "bimodal." While the two frequencies are not identical, for all practical purposes we should recognize that there are two "modal" categories of ratings.

Percentile Bars

-Percentile bars visually represent distributions -skewed with a tail to the right -mean to the right of the median -distance between p75 and p50 greater than distance between p25 and p50 ex. 8.12

Shapes of Distributions

Distributions can take many shapes. We can better understand relationships among data by visualizing their distribution. Distribution shapes give a visual representation of how the distribution is "centered" and how the data points are dispersed.

Example - Survey Data Pg. 8.4

In this example, there are two different sets of data to be compared. The current graph represents survey data. We should not be surprised to find that it has considerable variation and is somewhat skewed. Notice the positional relationship of the mean to the bulk of the data.

Example - Both Data Sets Pg. 8.6

It is not unusual for a picture to look like this, where the survey data are wider and higher and your own company data are closer together.

The Normal Distribution -2/+2

Notice that the area between -2 standard deviations and +2 standard deviations is now 95.45%. When we have a perfectly normal distribution, the area between -2 and +2 standard deviations increases from "at least 75%" to "precisely 95.45%." +/- 3 is 99.73%

Characteristics of Nonsymmetric (Skewed) Distributions -Left:Example

The distribution of test results for a group of participants enrolled in a T3 seminar (or other WorldatWork certification classes), after they have completed the seminar, is typically negatively skewed, with far more individuals earning high scores than individuals earning low scores

Characteristics of Nonsymmetric (Skewed) Distributions -Left

distributions skewed with a tail to the left, the mean has a smaller numerical value than the median, which in turn has a smaller numerical value than the mode. We refer to this as a negatively skewed distribution. In such distributions, the most extreme "outliers" typically have negative z-scores.

Characteristics of Nonsymmetric (Skewed) Distributions Generally Speaking: Examples

- The number of factors typically used when making executive compensation decisions by organizations is generally positively skewed. Some organizations might use only one factor, while others might use 2-6 factors. Some may go 7 or more. -The distribution of ages for employees in the IT sector is generally skewed with a tail to the right. -The distribution of test scores for a set of students who have not studied at all for an exam is generally positively skewed. -The distribution of pay for major league baseball players is typically skewed with a tail to the right.

Distribution

-If the distribution is perfectly symmetric with one and only one mode, the mean, median, and mode will be equal. If the distribution is perfectly symmetric and there is more than one mode (e.g., bimodal), the mean and median will be equal to each other but not to the mode. -A flat distribution has no mode but is not necessarily symmetric. -If mean, median and mode are equal, this does not mean the distribution will necessarily be normal.

Characteristics of Nonsymmetric (Skewed) Distributions

-has one side not balanced against the other side. Thus, the distribution is said to be skewed. Distributions may be skewed with a tail to the left or skewed with a tail to the right. The salary distributions for most organizations are skewed to the right.

Summarizing the Normal Distribution

1. Approximates the distribution of naturally occurring phenomena - Most data, including compensation data, are not perfectly normal. We sometimes hope that our data are approximately normal, but this should not be our goal. 2. An essential tool sometimes misused - The normal distribution is an essential tool for inferential statistics that is sometimes misused to force a desired outcome. 3. Not perfectly normal - Most data, including compensation and benefits data, are not perfectly normal.

Shapes of Distributions: Questions to ask

1. Is the shape symmetrical? (Is one side a mirror image of the other side?) 2. Is it tall and skinny? 3. Is it short and squatty? 4. Is it bell-shaped? -If it is not symmetrical, then it is skewed (with a tail trailing off in one direction). -The visual characteristics of distributions have corresponding mathematical values

Normal Distribution quick fact

The normal distribution does not apply to things that do not naturally occur. For example, performance ratings would not apply.

Example - Company Data Pg. 8.5

This graph represents a company's data and can be compared to the previous graph taken from survey data. We should not be surprised to find that it has considerably less variation than the survey data. In this case, it is also much more symmetric. However, we do not want to conclude that it is "normal" (bell-shaped).

Characteristics of Nonsymmetric (Skewed) Distributions Generally Speaking:

the mode, median and mean follow a particular order in skewed distributions. In distributions skewed with a tail to the right, the mean typically has a bigger numerical value than the median, which in turn has a bigger numerical value than the mode. We refer to this as a positively skewed distribution. In such distributions, the most extreme "outliers" typically have positive z-scores.