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Variability Solutions
- The simplest way is to calculate the range , which is simply:
range = highest score - lowest score + 1 Here, range = 5 - 1 + 1 = 5
Compare that scrawny range to the range of these scores:
X = 0, 2, 55, 6422 range = 6422 - 0 + 1 = 6,423! MORE VARIABILITY!
- Eyeballing, I would say the second group.
Look at them this way: Mathematically:
0
1
2
3
4
How many in each
group?
2 3 4 Group
0
1
2
3
4
How many in each
group?
1 2 3 4 5 Group
NEXT: Calculate deviations about the mean:
X (X - μ) X (X - μ) 2 2 - 3 = -1 1 1 - 3 = - 3 3 - 3 = 0 2 2 - 3 = - 3 3 - 3 = 0 3 3 - 3 = 0 3 3 - 3 = 0 3 3 - 3 = 0 3 3 - 3 = 0 4 4 - 3 = 1 4 4 - 3 = 1 5 5 - 3 = 2 μ = 3 Σ(X - μ) = 0! μ = 3 Σ(X - μ) = 0!
The sum of the deviations added up to 0 both times! But we knew that from our balancing point lecture: Σ(X - μ) = 0 OR Σ(X - ) = 0 ALWAYS!
MORE VARIABILITY!
Range 1 = 4 - 2 + 1 = 3
Range 2 = 5 - 1 + 1 = 5
Since the deviations always add up to 0, we can’t simply add them up and divide by how many there are to get an average.
So, we’d best square them…
STEP #1 STEP #2 STEP #3 STEP #1 STEP #2 STEP # X (X - μ) (X - μ) 2 X (X - μ) (X - μ) 2 2 2 - 3 = -1 (-1)(-1) = 1 1 1 - 3 = -2 (-2)(-2) = 4 3 3 - 3 = 0 (0)(0) = 0 2 2 - 3 = -1 (-1)(-1) = 1 3 3 - 3 = 0 (0)(0) = 0 3 3 - 3 = 0 (0)(0) = 0 3 3 - 3 = 0 (0)(0) = 0 3 3 - 3 = 0 (0)(0) = 0 3 3 - 3 = 0 (0)(0) = 0 4 4 - 3 = 1 (1)(1) = 1 4 4 - 3 = 1 (1)(1) = 1 5 5 - 3 = 2 (2)(2) = 4 μ = 3 Σ(X - μ) = 0 Σ(X - μ) 2 = 2 μ = 3 Σ(X - μ) = 0 Σ(X - μ) 2 = 10
THESES ARE THE
SUMS OF SQUARES!
“ SS ”
STEP
σ^2 = 2/6 = 0.33 σ^2 = 10/6 = 1.
STEP
σ = σ =
DONE!
- It is the square root of the average squared deviation.
- It is the average distance scores are from their mean.
- A) X (X - μ) (X - μ) 2 Range = 8 - 4 + 1 = 5
σ^2 (variance) = Σ(X – μ) 2 OR σ^2 = SS N N
σ (standard deviation) = OR σ =
σ SS/N
D) X (X - μ) (X - μ) 2 N = 6 2 -9.83 96. 9 -2.83 8. 11 -0.83 0. 15 3.17 10. 17 5.17 26. 17 5.17 26. μ = 11.83 Σ(X - μ) = 0 ***** Σ(X - μ) 2 = 168.84 (SS)
σ2 = 168.84/6 =28.
σ =
- D wasn’t so fun, was it? As soon as you have a mean that is something ugly like 11.83, you have a lot of work ahead of you! Also, N = 6 is manageable, but what if N = 56?!
Aha! That’s the beauty of COMPUTATIONAL formulas! Try A-D again, using:
RATHER THAN
Redo them and check your answers!
A) X X^2
ΣX = 24 ΣX^2 = 152
B) X X^2
***** Σ(X - μ) won’t equal exactly zero due to rounding error.
SS = ΣX^2 – (ΣX) 2
N
SS = Σ(X - μ) 2
SS = 152 – (24) 2 = 152 – 576 = 152 – 144 = 8
σ^2 = 8/4 = 2
σ =
SS = 609 – (55) 2 = 609 – 3025 = 609 – 605 = 4
σ^2 = 4/5 = 0.
σ =
ΣX = 55 ΣX^2 = 609
C) X X^2
ΣX = 80 ΣX^2 = 1942
D) X X^2
ΣX = 71 ΣX^2 = 1009
This way was much easier than using the conceptual formula!
BUT ALSO NOTICE THAT YOUR STAT CALCULATORS WILL
CALCULATE ΣX, ΣX^2 , AND σ AFTER YOU PUT THE DATA IN JUST ONCE! WHOOPEE! $10 WELL SPENT, I’D SAY! NOW YOU CAN CHECK YOUR WORK!
SS = 1942 – (80) 2 = 1942 – 6400 = 1942 – 1280 = 662
σ^2 = 662/5 = 132.
σ =
SS = 1009 – (71) 2 = 1009 – 5041 = 1009 – 840.17 = 168.
σ^2 = 168.83/6 = 28.
σ =
NOTICE:
Our answer is not the exactly the same as before. This is simply rounding error. DON’T SWEAT IT!
