Variance and Standard Deviation
Variance The rst rst important number describing a probability distribution is the mean or expected value E(X) The next one is the variance Var(X) = ˙2(X) The square root of the variance ˙is called the Standard Deviation If f(x i) is the probability distribution function for a random variable with range fx 1;x 2;x 3;:::gand mean = E(X) then:
Chapter 4 Variances and covariances
many distributions the simplest measure to calculate is the variance (or, more precisely, the square root of the variance) De nition The variance of a random variable X with expected value EX = is de ned as var(X) = E (X )2 The square root of the variance of a random variable is called its standard deviation, sometimes denoted by sd(X)
Chapter 4 Variances and covariances
culate for many distributions is the variance There is an enormous body of probability †variance literature that deals with approximations to distributions, and bounds for probabilities and expectations, expressible in terms of expected values and variances Definition Thevariance of a random variable X with expected valueEX D„X is
Variance, covariance, correlation, moment-generating functions
– Notes: In contrast to expectation and variance, which are numerical constants associated with a random variable, a moment-generating function is a function in the usual (one-variable) sense (see the above examples) A moment generating function characterizes a distribution uniquely, and
3 Variances and covariances - Statistics
An important summary of the distribution of a quantitative random variable is the variance This is a measure how far the values tend to be from the mean The variance is defined to be var(X) = E(X −EX)2 The variance is the average squared difference between the random variable and its expec-tation
Lecture 9: Variance, Covariance, Correlation Coefficient
beamer-tu-logo Variance CovarianceCorrelation coefficient Lecture 9: Variance, Covariance, Correlation Coefficient Kateˇrina Sta nkovᡠStatistics (MAT1003)
Lecture 20, Expectation & Variance II
Lecture 20, Expectation & Variance II Example 1 The expected number of successes when n independent Bernouli trials are performed, where p is the probability of success on each trail, is np We discuss two approaches One using algebraic operation and the other one relies on the linear property of expectations Approach #1: E(X) = Xn k=1 kp(X
Pooled Variance t Test - gimmenotes
(Variance) F Among (Methods) 4 - 1 = 3 348 116 11 6 Within (Error) 12 - 4 = 8 80 10 Total 12 - 1 = 11 428 Source of Variation Degrees of Freedom Sum of Squares Mean Square (Variance) F Among (Methods) 4 - 1 = 3 348 116 11 6 Within (Error) 12 - 4 = 8 80 10 Total 12 - 1 = 11 428
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Pooled Variance
t TestTests means of 2 independent populations having
equalvariancesParametric test procedure
Assumptions
Both populations are normally distributed
If not normal, can be approximated by normal distribution (n130 & n230 )Population variances are unknownbut assumed equal
Two Independent Populations
Examples
An economist wishes to determine whether
there is a difference in mean family income for households in 2 socioeconomic groups.An admissions officer of a small liberal arts
college wants to compare the mean SAT scores of applicants educated in rural high schools & in urban high schools.Pooled Variance t Test Example
to see if there a difference in dividend yield between stocks listed on the NYSE & NASDAQ.NYSENASDAQ
Number2125
Mean3.272.53
Std Dev1.301.16
Assuming equalvariances, is
there a difference in average yield (= .05)?© 1984-1994 T/Maker Co.
Pooled Variance t Test
Solution
H0:1 -2= 0 (1 = 2)
H1:1 -20 (1 2)
.05 df 21 + 25 -2 = 44Critical Value(s):
Test Statistic:
Decision:
Conclusion:
t02.0154-2.0154 .025Reject H0Reject H0
.025 t02.0154-2.0154 .025Reject H0Reject H0
.025 2.03 251 21
11.510
2.533.27t
Reject at
= .05There is evidence of a
difference in meansTest Statistic
Solution
tXX Snn SnSnS nn P PFHGIKJ
F H I K 12122 12 211
2 22
2 12 22
11
3272530
1510121
1 25
203
11 11
211130251116
2112511510
chafafaf afafafaf afafafafafaf tXX Snn SnSnS nn P PFHGIKJ
F H I K 12122 12 211
2 22
2 12 22
11
3272530
1510121
1 25
203
11 11
211130251116
2112511510
chafafaf afafafaf afafafafafaf equalvariances, is there a difference in the average miles per gallon (mpg) of two car models (= .05)?You collect the following:
SedanVan
Number1511
Mean22.0020.27
Std Dev4.773.64
Pooled Variance t Test
Thinking Challenge
Alone Group ClassTest Statistic
Solution*
tXX Snn SnSnS nn P PFHGIKJ
F H I K 12122 12 211
2 22
2 12 22
11
220020270
187931
15 1 11 10011 11
151477111364
15111118793
chafafaf afafafaf afafafafafaf tXX Snn SnSnS nn P PFHGIKJ
F H I K 12122 12 211
2 22
2 12 22
11
220020270
187931
15 1 11 10011 11