Standard Deviation MDM4U: Mathematics of Data Management A deviation - PDF document

s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Standard Deviation MDM4U: Mathematics of Data Management A deviation is the difference between any value in a data set and the mean. For a population, a deviation is x − µ , while for a sample, it is x − x . Measures of Spread (Part 2) A data set with larger deviations has a greater spread. Standard Deviation and z -Scores Values less than the mean have negative deviations, while those above the mean have positive deviations. MDM4U: Data Management The most common measure of deviation within a data set is the standard deviation , which measures the average distance of a datum from the mean of the data set. MDM4U: Data Management — Measures of Spread (Part 2) Slide 1/15 Slide 2/15 s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Standard Deviation Standard Deviation Standard Deviation of a Population Example Calculate the standard deviation for the following data. �� ( x − µ ) 2 σ = 5 7 7 8 10 14 19 N Solution: Calculate the mean of the data. Since a sample tends to underestimate the deviations in a x = 5 + 7 + 7 + 8 + 10 + 14 + 19 population, the formula is slightly different for samples. = 10. 7 Standard Deviation of a Sample Make a table, with columns for x , x − x , and ( x − x ) 2 . �� ( x − x ) 2 s = n − 1 MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 3/15 Slide 4/15 s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Standard Deviation Standard Deviation ( x − x ) 2 Datum x x − x There is a faster method of computing the standard 5 − 5 25 x 1 deviation, developed prior to the emergence of statistical x 2 7 − 3 9 software. 7 − 3 9 x 3 This computational formula deals with the squares of each x 4 8 − 2 4 datum, rather than any differences from the mean. x 5 10 0 0 x 6 14 4 16 Computational Formula for Standard Deviation (Sample) x 7 19 9 81 �� x 2 − nx 2 � ( x − x ) 2 = 144 s = n − 1 √ � 144 Therefore, s = 7 − 1 = 24 ≈ 4 . 899. MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 5/15 Slide 6/15

s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Standard Deviation Variance Example Variance in a data set is a measure of dispersion of the data. Verify the computational formula using the earlier data. Mathematically, variance is the square of the standard deviation. x 2 Datum x Variance of a Population 5 25 x 1 x 2 7 49 � ( x − µ ) 2 σ 2 = 7 49 x 3 N x 4 8 64 10 100 Variance of a Sample x 5 x 6 14 196 � ( x − x ) 2 s 2 = 19 361 x 7 � x 2 = 844 n − 1 � 844 − 7(10) 2 Therefore, s = ≈ 4 . 899. 7 − 1 MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 7/15 Slide 8/15 s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Variance Variance Example Your Turn Calculate the variance and standard deviation of the Calculate the variance of the earlier data. following data. √ Since the standard deviation is s = 24, the variance is 3 7 9 10 13 24 s 2 = 24. Solution: The mean is x = 3 + 7 + 9 + 10 + 13 + 24 Note that the variance is always calculated as part of the = 11. 6 process of calculating the standard deviation. Use the computational formula to calculate the variance and Also note that for both the standard deviation and the standard deviation. variance, we will almost always be using the formula for a sample, since we do not often have data for the entire population. MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 9/15 Slide 10/15 s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e Variance z -Scores x 2 Datum x A z-score measures the number of standard deviations a 3 9 x 1 datum is from the mean. x 2 7 49 z -Score for a Population 9 81 x 3 z = x − µ x 4 10 100 x 5 13 169 σ x 6 24 576 � x 2 = 984 z -Score for a Sample z = x − x Therefore, the variance is s 2 = 984 − 6(11) 2 = 258 5 = 51 . 6. s 6 − 1 � 258 The standard devtaion is s = 5 ≈ 7 . 183. A negative z -score indicates a datum is below the mean, while a positive z -scores indicates it is above. MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 11/15 Slide 12/15

s t a t i s t i c s o f o n e v a r i a b l e s t a t i s t i c s o f o n e v a r i a b l e z -Scores z -Scores Example The second datum is below the mean, so its z -score will be negative. A data set has a mean of 5 and a standard deviation of 1 . 2. z = 3 − 5 = − 2 1 . 2 = − 5 Determine the z -scores for data with values of 6 . 2 and 3. 3. The datum is one-and-two-thirds 1 . 2 standard deviations below the mean. Solution: The first datum is above the mean, so its z -score will be positive. z -scores will play a very important role in the last unit of this The datum is z = 6 . 2 − 5 = 1 . 2 course when we deal with continuous probability distributions. 1 . 2 = 1 standard deviation 1 . 2 above the mean. MDM4U: Data Management — Measures of Spread (Part 2) MDM4U: Data Management — Measures of Spread (Part 2) Slide 13/15 Slide 14/15 s t a t i s t i c s o f o n e v a r i a b l e Questions? MDM4U: Data Management — Measures of Spread (Part 2) Slide 15/15