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Notes: Standard Deviation and the Normal Model
Standard deviation is a measure of spread, or variability. The smaller the standard deviation, the less variability is present in the data. The larger the standard deviation, the more variability is present in the data.
Standard deviation can be used as a ruler for measuring how an individual compares to a group.
To measure how far above or below the mean any given data value is, we find its standardized value , or z-score.
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To standardize a value, subtract the mean and divide by the standard deviation.
Measure your height in inches. Calculate the standardized value for your height given that the average height for women is 64.5 inches with a standard deviation of 2.5 inches and for men is 69 inches with a standard deviation of 2.5 inches. Are you tall?
z height
Suppose the average woman’s shoe size is 8.25 with a standard deviation 1.15 and the average male shoe size is 10 with a standard deviation of 1.5. Do you have big feet?
z shoe
Suppose Sharon wears a size 9 shoe and Andrew wears a size 9. Does Sharon have big feet? Does Andrew?
z Sharon zAndrew
In order to compare values that are measured using different scales, you must first standardize the values. The standardized values have no units and are called z-scores. Z-scores represent how far the value is above the mean (if positive ) or below the mean (if negative ).
Ex: z = 1 means the value is one standard deviation above the mean z = -0.5 means the value is one-half of a standard deviation below the mean
The larger the z-score, the more unusual it is.
Standardized values, because they have no units, are therefore useful when comparing values that are measured on different scales , with different units , or from different populations.
Adding a constant to all of the values in a set of data adds the same constant to the measures of center and percentiles. It does not, however, affect the spread.
Example : Add 5 to each value in the given set of data (on the left) to form a new set of data (on the right). Then find the indicated measures of center and spread.
Center: Center:
x = x =
M = M =
Mode = Mode = Spread: Spread: Range = Range = IQR = IQR = SD = SD = Multiplying a constant to all of the values in a set of data multiplies the same constant to the measures of center and spread.
Example : Multiply each value in the given set of data (on the left) by 2 to form a new set of data (on the right). Then find the indicated measures of center and spread.
Center: Center:
x = x =
M = M =
Mode = Mode = Spread: Spread: Range = Range = IQR = IQR = SD = SD =
By standardizing values, we shift the distribution so that the mean is 0 , and rescale it so that the standard deviation is 1. Standardizing does not change the shape of the distribution.
The Normal model: is symmetric and bell-shaped. follows the 68-95-99.7 Rule o About 68% of the values fall within one standard deviation of the mean. o About 95% of the values fall within two standard deviations of the mean. o About 99.7% (almost all) of the values fall within three standard deviations of the mean.
The larger sigma, the more spread out the normal model appears. The inflection points occur a distance of sigma on either side of mu.
To standardize Normal data, subtract the mean ( mu ) and divide by the standard deviation ( sigma ).
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To assess normality: Examine the shape of the histogram or stem-and-leaf plot. A normal model is symmetric about the mean and bell-shaped. Compare the mean and median. In a Normal model, the mean and median are equal. Verify that the 68-95-99.7 Rule holds. Construct a normal probability plot. If the graph is linear, the model is Normal.
Nearly Normal:
Skewed distribution:
