# Numerical Measures of Variability Deviations and the Spread of Data A deviation is the difference between an observation value and the mean of its sample.

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Numerical Measures of Variability

Deviations and the Spread of Data A deviation is the difference between an observation value and the mean of its sample or population distribution. The sum total of all deviations in a sample or population always equals zero. But deviations tell us something more about the sample or population than does a measure of central tendency, such as the mean. The greater the deviations – both positive and negative – the more spread out is the data.

Example: Consider the difference between the ages of students in a KKU school bus….. …and the ages of passengers in a typical city bus. Lets take a sample of the ages of n = 20 passengers (observations) in each of the two buses. Note that, even though the spread of ages on the city bus is larger than that of the school bus, we cant use the total deviations to show this! *Mean = Σtotal/n = Σtotal/20

Measures of Spread or Variability RangeInterquartile RangeMean Absolute DeviationVariance or Standard Deviation ******Coefficient of Variation

The Range = Largest Value – Smallest Value

Returning to the bus example the range R on the two buses is R(KKU) = 24 – 19 = 5 and R(City) = 67 – 2 = 65

To use more of the information in the data set, the Interquartile Range can be used instead. The Interquartile Range is found by ordering the data set as above, eliminating the lowest quarter (25%) of the data set and the highest quarter (25%) of the data set, and then finding the range of the middle half (50%) of the data set.

Back to deviations to measure spread Deviations measure spread, but… Always sum total to zero So we cannot use the sum total as a measure of spread If all deviations could be made positive, then deviations could be used. But how? Take the absolute value of the deviations (ignore the minus signs) Square the deviations (all squared 2 values are positive!)

Mean Absolute Deviation (MAD) In mathematics, a numbers absolute value is that same number, but expressed as a positive number without the minus sign (if it is a negative number). Straight vertical lines denote the absolute value. Absolute value of 3 = |3| = 3 Absolute value of – 3 = |-3| = 3 The Mean Absolute Deviation (MAD) takes the absolute value of each of the sample deviations, and then gets the mean of these values for the sample as a whole.

Because there are 7 sample values, the MAD = 32/7 = 4.57 Just ignore the minus signs!

Returning to the School Bus & City Bus Example….. Note that the City Buss MAD is much larger than the KKU School Buss MAD, just as our intuition would suggest. n = 20 passengers MAD = Σ|Deviation|/n = Σ|Deviation|/20

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