MEAN ABSOLUTE DEVIATION WarmUp Problems Find the mean

MEAN ABSOLUTE DEVIATION

Warm-Up Problems: Find the mean, median, mode and range of the following data set: {9, 7, 7, 4, 5, 7}

Objective: By the end of this lesson, you will be able to: find the standard deviation of a data set find the mean absolute deviation of a data set

In statistics, the measure of central tendency gives a single value that represents the whole value. But the central tendency cannot describe the observation fully. The measure of dispersion helps us to study the variability of the items.

Two formulas which find the dispersion of data about the mean are: standard deviation – squares each difference from the mean to eliminate the negative differences. mean absolute deviation – uses absolute value of each difference from the mean to eliminate the negative differences.

Standard Deviation Standard deviation is a number used to tell how measurements for a group are spread out from the average (mean), or expected value. A low standard deviation means that most of the numbers are very close to the average. A high standard deviation means that the numbers are spread out.

Standard deviation is used in a variety of fields and professions. Often, the standard deviation is used as a measurement of how accurate or reliable a set of data might be. In a scientific study, the standard deviation of the data can indicate whether the data was consistent and therefore can help scientists consider how solid any scientific conclusion based on their data might be. In meteorology, the standard deviation of weather data can help an observer understand how reliable or predictable a certain weather forecast will be. In predicting the movement of the stock market and in determining how profitably or reliably an investment can be made to function. If there is a large amount of disparity between a stock's value or profit at different times, that could indicate that the stock is volatile; if there is a low standard deviation, that could indicate that a stock is solid and would make a safe investment.

Example of Standard Deviation The owner of the Chez Tahoe restaurant is interested in how much people spend at the restaurant. He examines 10 randomly selected receipts for parties of four and writes down the following data. 44, 50, 38, 96, 42, 47, 40, 39, 46, 50 He calculated the mean by adding and dividing by 10 to get x = 49. 2

Continued… Below is the table for getting the standard deviation: x x - 49. 2 (x - 49. 2 )2 44 -5. 2 27. 04 50 0. 8 0. 64 38 11. 2 125. 44 96 46. 8 2190. 24 42 -7. 2 51. 84 47 -2. 2 4. 84 40 -9. 2 84. 64 39 -10. 2 104. 04 46 -3. 2 10. 24 50 0. 8 0. 64 Tot al 2600. 4

Continued… We first find the variance: 2600. 4 = 288. 7 10 – 1 The standard deviation is the square root of 289 = 17. Since the standard deviation can be thought of measuring how far the data values lie from the mean, we take the mean and move one standard deviation in either direction. The mean for this example was about 49. 2 and the standard deviation was 17. We have: 49. 2 - 17 = 32. 2 and 49. 2 + 17 = 66. 2 What this means is that most of the patrons probably spend between $32. 20 and $66. 20.

Mean Absolute Deviation, referred to as MAD, is a better measure of dispersion than the standard deviation when there are outliers in the data. An outlier is a data point which is far removed in value from the others in the data set. It is an unusually large or an unusually small value compared to the others.

Mean Absolute Deviation mean absolute deviation (MAD) The mean absolute deviation (MAD) of a data set is the average of the absolute values of all deviations from the mean in that set.

The Mean Absolute Deviation is calculated in five simple steps. 1) Determine the Mean: Add all numbers and divide by the count 2) Determine deviation of each variable from the Mean: Subtract the mean from each number 3) Make the deviation 'absolute' by squaring and determining the roots (i. e. eliminate the negative aspect) 4) Find the sum of the absolute values 5) Divide the sum by the number of data items

Mean Absolute Deviation For example, in the data set: {1, 2, 3, 4, 5}, the mean is 3. The deviations from the mean are {-2, -1, 0, 1, 2}. i. e. 1 – 3, 2 – 3, 3 – 3, etc. The absolute deviations from the mean are {|-2|, |-1|, |0|, |1|, |2|}. The MAD is (2 + 1 + 0 + 1 + 2) / 5 = 1. 2. The MAD is a measure of, on average, how far the values in a data set are from the mean.

Mean Absolute Deviation What does a low mean deviation means? A low standard deviation would mean that there is not much variation from the mean value.

Find the mean absolute deviation Test scores for 6 students were : 85, 92, 88, 80, 91 and 20. 1. Find the mean: (85+92+88+80+91+20)/6=76 2. Find the deviation from the mean: 85 -76=9 92 -76=16 88 -76=12 80 -76=4 91 -76=15 20 -76=-56

3. Find the absolute value of each deviation from the mean:

4. Find the sum of the absolute values: 9 + 16 + 12 + 4 + 15 + 56 = 112 5. Divide the sum by the number of data items: 112/6 = 18. 7 The mean absolute deviation is 18. 7.

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