Examples:Look for the words "normally distributed" in a question before referring to the Normal Distribution Standard Deviation chart seen on this page. When using the chart, your information should fall on the increments of one-half of one standard deviation as shown in the chart.
1.At the New Age Information Corporation, the ages of all new employees hired during the last 5 years are normally distributed. Within this curve, 95.4% of the ages, centered about the mean, are between 24.6 and 37.4 years. Find the mean age and the standard deviation of the data. /Solution:As was seen in Example 1, 95.4% implies a span of 2 standard deviations from the mean. The mean age is symmetrically located between -2 standard deviations (24.6) and +2 standard deviations (37.4).
The mean age is years of age.
From 31 to 37.4 (a distance of 6.4 years) is 2 standard deviations. Therefore, 1 standard deviation is (6.4)/2 = 3.2 years. /
2.The amount of time that Carlos plays video games in any given week is normally distributed. If Carlos plays video games an average of 15 hours per week, with a standard deviation of 3 hours, what is the probability of Carlos playing video games between 15 and 18 hours a week? /
Solution: The average (mean) is 15 hours. If the standard deviation is 3, the interval between 15 and 18 hours is one standard deviation above the mean, which gives a probability of 34.1% or 0.341, as seen in the chart at the top of this page. /
Data can be "skewed", meaning it tends to have a long tail on one side or the other:
Negative Skew / No Skew / Positive Skew/ Negative Skew?
Why is it called negative skew? Because the long "tail" is on the negative side of the peak.
People sometimes say it is "skewed to the left" (the long tail is on the left hand side)
The mean is also on the left of the peak.
The Normal Distribution has No Skew
A Normal Distribution is not skewed.
It is perfectly symmetrical.
And the Mean is exactly at the peak. /
Positive Skew
And positive skew is when the long tail is on the positive side of the peak, and some people say it is "skewed to the right".
The mean is on the right of the peak value. /
MORE ON Skewness
Skewness is a measure of degree of asymmetry of the distribution.
1.Symmetric
Mean, median, and mode are all the same here; mound shaped, no skewness (symmetric).
The above distribution is symmetric.
2.Skewed Left
Mean to the left of the median, long tail on the left.
The above distribution is skewed to the left.
3.Skewed Right
Mean to the right of the median, long tail on the right.
The above distribution is skewed to the right.
When one has very skewed data, it is better to use the median as measure of central tendency since the median is not much affected by extreme values.
/ Example: Income DistributionHere is some data I extracted from a recent Census.
As you can see it is positively skewed ... in fact the tail continues way past $100,000