Chapter 4
Standard Deviation
It is the square root of the variance. Standard deviation (s) is computed by extracting the square root of the variance. -Example: Test score of 5 students (n) is given below: 85, 71, 91, 76, 82 (with mean = 81) 1. Subtract each test score from the mean 2. Square each result 3. Once each result it squared, add them up and get the total 4. Subtract the total number of test scores by 1 5. Divide the total you got in step 3 by the total you got in step 4 6. Then take the square root
Mean Deviation (MD)
MD is the average deviation of all observations from the mean. -Example: Test score of 5 students (n) is given below: 85, 71, 91, 76, 82 1. Find the mean 2. Subtract each test score from the mean 3. Take the absolute value of each result 4. Add the results up and divide by the total number of test scores
Mean
Mean (X-bar) is computed by summing all the observations in the sample and dividing the sum by the number of observations. -The mean considers the magnitude of each observation -The mean is affected by the value of each observation of the distribution
Median
Median is the observation in rank order that divides the distribution into equal parts. -The midpoint of the observation -The 50th percentile of a set of numbers -The median is not affected by extreme values -For an even number of observations, the median is the average of the two middle values
Coefficient of Variation (CV)
One important implication of the mean and the standard deviation is the CV. -Depicts the size of the standard deviation relative to its mean -Free of measurement units -Able to compare relative variation of unrelated variables -The formula of CV is: (Answer should be a %) 100 * s/X Bar (Mean)
Range
Range is the difference in value between the highest (maximum) and lowest (minimum) observation. Range considers extreme values only.
Properties of Standard Deviation
Standard deviation is best used for normal (symmetrical) distribution with a single peak. When data are normally distributed, SD divides the observations roughly as: 68% of values within 1 SD 95% of values within 2 SDs 99% of values within 3 SDs
Mode
The observation that occurs most frequently. -There may be several modes if several values occur with the same frequency -There may be Primary and Secondary modes if values cluster in several places but with unequal frequencies
Central Tendency
The tendency of a set of data to center around certain numerical values. -The center of the distribution where values cluster -3 most commonly used values are: Mean Median Mode
Variation
Used to describe variability. Variability refers to the spread or dispersion of data around the measures of central tendency. -Measures of central tendency without measures of variability may result in misinterpretation of data -Data sets may have the same mean but different variability
Measures of Variation
Variance (s2) is the average squared deviations of the observations from the mean -s2, however, is not usually used as descriptive statistics Standard deviation (s) is the most widely used measure of variation of all other choices -s is the square root of the variance -Larger the s, the more heterogeneous the distribution (smaller the s, the more homogeneous the distribution)
Median - Pros & Cons
-Gives a typical observation as the 50th percentile -Better indicator for skewed observations -Not as widely used as mean
Mode - Pros & Cons
-Gives the most popular value -Attention to clustering -Not as widely used as mean
Mean - Pros & Cons
-Most widely used -Best used if data are normally distributed -Many business & commercial applications -Not good for skewed distributions and affected by extreme values -Not good in very small samples
Central Tendency
-When large samples are randomly drawn from a population, the means of these samples should be normally distributed (n should be at least 25). -Larger the sample size, the better in general
Asymmetrical Distribution
...
Symmetrical Distribution
...
Measures of Variation
3 most commonly used variations are: Range Mean deviation Standard deviation
