How To Calculate Variance And Standard Deviation

Transcription

1 Variance and Standard Deviation Created 2 By Michael Worthington Elizabeth City State University Variance = ( mean ) 2 The mean ( average) is between the largest and the least observations Subtracting the mean from the observation value () will give a positive number if the observation is more than the mean But if the observation is less than the mean, the result will be a negative number So to compute the Variance, first square the differences to convert the results to ALL positive numbers, and total the squared values Standard Deviation = square root of Variance Symbols of a variable is generally indicated by placing a bar over the variable = mean of ( mean ) σ 2 = Variance of a column of values σ = Standard Deviation of the values

2 Mean of The values (observations) are provided by problems, so the first step is to compute the mean - mean (- m ) 2 Total = 3 Arithmetic Mean (average) of = the total divided by the number (n) of observations: 3 n = Difference between value and After listing the mean, then find the differences mean - mean -= -= -= -= + -= - (- m ) 2 Total = Since the mean is the middle value, then the total of the differences must be zero (or very close) Square the Difference Square the Differences to Make Positive, and the total of the column is the mean - mean + - (- m ) 2 Variance (σ 2 ) = Total Variance (σ 2 ) measures the dispersion of observation values around the mean 2

3 Standard mean - mean + - (- m ) 2 Variance (σ 2 ) = The standard deviation = the square root of the variance = = 3.2 Compute the mean - mean (- m ) 2 2 Total = Click on the mean of the values 2 Total the Third Column mean - mean (- m ) Click on the total of the ( - mean ) column 3

4 Compute Standard Deviation 2 mean - mean - + (- m ) 2 Click on the Standard Deviation,.32 Variance (σ 2 ) = Total Normal Distribution Frequently graphs of data form a bell-shaped curve, which is called a distribution Height = frequency of the value The tails taper off, but never quite touch the axis (horizontal line) Tangent Lines Tangent Lines only touch one point on the

5 of Tangent Lines Horizontal Line has zero slope Rising Line has positive slope Falling Line has negative slope Upward slope is positive, and downward is negative One Normal distributions have only one hump like an Arabian camel Normal Distribution The sign of the slopes of lines tangent to a normal distribution curve only changes once -- it rises (positive slope), becomes horizontal (zero slope), then it falls (negative slope) Rule Also called the Normal Rule is the general rule for interpreting a normal curve Mean % Std Dev Std Dev % of the observed data will fall within one standard deviation to either side of the mean

6 Empirical Rule % of the observed data will fall within two standard to either side of the mean Mean % 2 Std Dev 2 Std Dev.% of the observed data will fall within three standard deviations to either side of the mean Summary of Empirical % of the observed data will fall within ONE standard deviation to either side of the mean % of the observed data will fall within TWO standard deviations to either side of the mean.% of the observed data will fall within THREE standard deviations to either side of the mean Chebyshev s This formula applies to all data groups (not just normal distributions) to compute the percentage of observations that fall within K distributions % of observations = ( / k 2 )

7 Chebyshev s Example % of observations = ( / k 2 ) The percentage of data observations that fall within 2. standard deviations k = 2. ( / 2. 2 ) ¼ = -.2. or % Notice that this % is less than the % of the Empirical Rule because normal distributions are more concentrated Another Example of Chebyshev s % of observations = ( / k 2 ) The percentage of data observations that fall within 2. standard deviations k = 2. ( / 2. 2 ) ( /.2).. or % Comparison Empirical Rule only applies to Normal Distributions Chebyshev s Theorem applies to all types of distributions The END