Download Section 2.5 Measures of Variability is the degree to which the data

Section 2.5
Measures of Variability
___________________________________ is the degree to which the data are spread out.
We will discuss three numerical measures for describing the dispersion, or spread, of data:
____________________ , ___________________________, __________________________
1.
The __________________ is the largest measurement minus the smallest measurement.
Range =
(Benefits: easy to calculate, and easy to interpret; downsides: it is insensitive when the data set is large.)
2.
____________________________ measures how far a set of numbers is spread out.
The _________________________________________________for a sample of n measurements is equal to the sum of the
squared distances from the mean divided by n – 1.
Example: Calculate the sample variance for the following numbers: 2, 7, 3, 9, and 12.
Method 1
Method 2
Values
Mean
Deviation
Deviation Squared
x
(𝑥 − 𝑥̅ )
(𝑥 − 𝑥̅ )2
𝑥̅
2
Values
x
2
3
7
9
12
3
7
9
12
Values
Squared
𝑥2
Example: Calculate the variance.
Calculate the variance.
n = 40
∑𝑥 2 = 380
∑𝑥 = 100
3.
_______________________________ is a measure that is used to quantify the amount of variation or
dispersion of a set of data values
The _______________________________, s, for a sample of n measurements is equal to the square root of the
sample variance.
n
s  s2 
 (x  x )
2
i
i 1
n 1
Example: Calculate the sample standard deviation for the following numbers: 2, 9, 5, 7, 4, and 2.
Example: Calculate the variance and standard deviation.
n = 20
∑𝑥 2 = 18
∑𝑥 = 17
As before, Greek letters are used for populations and Roman letters for samples
The symbols:
Population Variance:_____,
sample variance ______
Population Standard Deviation ________,
sample standard deviation _____
The __________________ the standard deviation, the ______________ variable the data are.
The __________________ the standard deviation, the _______________ variation there is in the data.
Notice something very important: the unit of measurement (i.e.-ft., inche, miles, yrs, dollars,…) for standard deviation
will be the same units the original data was measured in and the same as the mean. Variance does not have the same
units. Its unit will be the square of the units for the original data.