Download measures of spread (deviation, standard deviation, variance)

Unit 5: Statistics of One Variable (Lesson 5 ~ Standard Deviation and Variance) The measures of spread or dispersion of a data set are quantities that indicate how closely a set of data clusters around its centre. There are several different measures of spread. A deviation is the difference between an individual value and the mean for the data. The larger the size of the deviations, the greater the spread in the data. (–) deviation values below mean values above (+) deviation NOTE: The sum of all the deviations of a set of data will equal zero!! For a SAMPLE: For a POPULATION: deviation = x i − x deviation = x i − µ The standard deviation is the square root of the mean of the squares of the deviations. For a SAMPLE: For a POPULATION: ∑ fi (xi − x)2 σ = ∑ fi (xi − µ)2 s=
n −1
N
NOTE: (n-­‐1) is used to compensate for the fact that a sample tends to underestimate the deviations of a population. For grouped data, where fi is the frequency for the given interval and mi is the midpoint of the interval, the standard deviation is calculated using: For a SAMPLE: For a POPULATION: s =
∑ fi (mi − x) 2 σ = ∑ fi (mi − µ) 2 N
n −1
The variance is the mean of the squares of the deviations (which is equal to the square of the standard deviation). For a SAMPLE: For a POPULATION: s
2
∑ fi ( x i − x )
=
n −1
2
σ
2
fi ( x i − µ) 2
∑
=
N
A z-­‐score is the number of standard deviations that a datum is from the mean. For a SAMPLE: For a POPULATION: x −x
x −µ
z = i
z = i
σ
s
values below (–) z-­‐score mean z = 0 values above (+) z-­‐score Example 1: Twenty students each took six shots with a basketball from the free throw line. The number of baskets made by each student was recorded. The results were as follows: 3 3 5 5 4 3 5 4 3 2 4 2 5 4 3 3 4 2 5 2 Determine the mean, standard deviation, variance, and z-­‐scores for the basketball data. Data Frequency Deviation Z-­‐score (xi) (fi) (xi -­‐ x ) (xi -­‐ x )2 fi(xi -­‐ x )2 x −x
z= i
s
TOTALS: