# Notes Mean, Variance, Standard Deviation, Covariance, Correlation

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```							                                Queens College, Economics 249, Geordan Hull, 1

Notes: Mean, Variance, Standard Deviation, Covariance, Correlation Coefficient

X values: 5, 10, 15
Y values: 10, 30, 50

   The first thing we need to do is to calculate the sample mean (sum of all the numbers divided
by the amount of numbers)
o X  (Sum of Xi)/n = (5+10+15)/3 = 30/3 = 10
o Y  (Sum of Yi)/n = (10+30+50)/3 = 90/3 = 30
 Use the numbers and means to construct the following table
(Xi-X bar)(Yi-Y
Xi      Yi      Xi-X bar      (Xi-X bar)2         Yi-Y bar (Yi-Y bar)2
bar)
2                             2
5      10       5-10=-5        (-5) =25         10-30=-20 (-20) =400            -5*-20=100
10      30      10-10=0           (0)2=0          30-30=0       (0)2=0              0*0=0
2                             2
15      50      15-10=5          (5) =25         50-30=20     (20) =400           5*20=100
5+10 10+3
400+0+400
+15= 0+50                       25+0+25=50                                      100+0+100=200
=800
30     =90

    To find the sample standard deviation for X, we need to find the sample variance for X
o Sample variance for X [Sum of (Xi-X bar)2]/n-1) = 50/(3-1) = 50/2 = 25
o Sample s.d. for X  square root of sample variance for X = 5
    To find the sample standard deviation for Y, we need to find the sample variance for Y
o Sample variance for Y [Sum of (Yi-Y bar)2]/n-1) = 8000/(3-1) = 800/2 = 400
o Sample s.d. for Y  square root of sample variance for Y = 20
    To find the sample covariance…
o [Sum of (Xi-X bar)(Yi-Y bar)]/(n-1) = 200/(3-1) = 200/2 = 100
    To find the sample correlation coefficient…
o Divide the sample covariance by (sample s.d. X * sample s.d. Y) = 100/(5*20) =
100/100 = 1

Keep In Mind

1. The highlighted values are the totals
2. Never get the sum of (Xi-X bar) or the sum of (Yi-Y bar). The sum of the deviation from
the mean will ALWAYS be equal to zero.
3. For the sum of (Xi-X bar)(Yi-Y bar), multiply each individual (Xi-X bar) and Sum of (Yi-
Y bar) together before adding them together.
4. You need to find the mean in order to calculate the variance or standard deviation.
5. You need to find the variance in order to calculate the correlation coefficient.
6. The larger the variance and/or standard deviation, the greater the variability.
7. The closer the correlation coefficient is to 0, the weaker the relationship. The closer the
correlation coefficient is to 1 (or –1), the stronger the relationship. For example, with a
correlation coefficient of 0.75, a 100% change in “A” will cause a 75% change in “B.”

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