Variability within Modeling Language Definitions - [email protected]
Measures of Variability - dooleykevin.comMeasures of Variability A single summary figure that...
Transcript of Measures of Variability - dooleykevin.comMeasures of Variability A single summary figure that...
Measures of Variability
A single summary figure that describes the spread of observations within a distribution.
DESCRIBING VARIABILITY
The amount by which scores are dispersed/spread/scattered in a distribution
0
2
4
6
Freq
uenc
y
FEAR
#1
0
2
4
6
Freq
uenc
y
FEAR
#2
0
2
4
6
Freq
uenc
y
FEAR
#3
0
2
4
6
Freq
uenc
y
FEAR
#4
Range
¨ Difference between the smallest and largest observations ¨ Pros and Cons
¤ Easy! J ¤ Values exist in the data set J ¤ Value depends on only two scores L ¤ Very sensitive to outliers L
¨ Examples: ¤ Fear scores: 1, 1, 5, 7, 9, 3, 1 ¤ Height
Deviations
¨ The average amount that a score deviates from the typical score. ¤ Score – Mean = Difference Score
¤ Average Mean Difference Score
nXX Σ
=
3515
554321
==++++
=X
0)( =Σ deviations
Score Score-Mean
Difference
1 1-3 -2
2 2-3 -1
3 3-3 0
4 4-3 1
5 5-3 2
*deviations always sum to zero To fix this, square each one…
Variance
¨ Mean of all squared deviation scores ¨ Steps
¤ 1. Calculate sample mean: ¤ 2. Calculate difference scores: score - mean ¤ 3. Square the difference scores (aka the Sum of Squares [SS]) ¤ 4. Add them up:
¤ 5. Take the average
Fear Score
Score -Mean
Difference Difference2
1 1-3 -2 4 2 2-3 -1 1 3 3-3 0 0 4 4-3 1 1 5 5-3 2 4
3515
554321
==++++
=X
10)( 2 =Σ deviations
2510)( 2
===∑N
deviationsAverage
2σ“sigma”
Variance: Definitional Formula
¨ Population ¨ Sample
NX∑ −
=2
2 )( µσ
1)( 2
2
−
−=∑
nXX
Syour old friend “sigma” …but lower case!
*Note the “n-1” in the sample formula! ** Degrees of freedom (df)
Symbol for sample variance
“mu”
Variance
¨ Use the definitional formula to calculate the variance.
44.4940
110)69()68()68()67()67()66()64()64()64()63(
)(
2
22222222222
22
==
−
−+−+−+−+−+−+−+−+−+−=
−=∑
S
S
nXX
S -1
Variance: Computational Formula
¨ Population ¨ Sample
2
222 )(
N
XXN∑ ∑−=σ
NX∑ −
=2
2 )( µσ 1
)( 22
−
−=∑
nXX
S
)1(
)( 22
2
−
−=∑ ∑
nnX
XS
Variance
¨ Use the computational formula to calculate the variance.
X X2
3 94 164 164 166 367 497 498 648 649 81
Sum: 60 Sum: 400)1(
)( 22
2
−
−=∑ ∑
nnX
XS
44.49360400910)60(400
2
2
2
2
=
−=
−=
S
S
S
Standard Deviation
¨ Rough measure of the average amount by which scores deviate on either side of the mean
¨ Steps: ¤ 1. Calculate variance (we just did this) ¤ 2. Take the square root
σ = σ 2
σ =(X − µ)∑N
2
s = s22
1)(
−
−= ∑
nXX
S
¨ Population ¨ Sample
Variability Example: Standard Deviation
)1(
)( 22
2
−
−=∑ ∑
nnX
XS
Mean: 6
Standard Deviation: 2.11
11.2940
110)69()68()68()67()67()66()64()64()64()63(
)(
2222222222
2
==
−
−+−+−+−+−+−+−+−+−+−=
−= ∑
S
S
nXX
S
11.244.49360400910)60(4002
=
=
−=
−=
SS
S
S
Practice!
¨ Calculate the range, variance, and standard deviation for the following set of “fear” scores
¨ Do this for the population AND the sample formulas
10, 8, 5, 0, 3, 4
Practice!
10, 8, 5, 0, 3, 4 ¨ Mean = 5 ¨ 10-5 = 5 à 25 ¨ 8-5 = 3 à 9 ¨ 5-5= 0 à 0 ¨ 0-5= -5 à 25 ¨ 3-5= -2 à 4 ¨ 4-5= -1 à 1 ¨ Sum of Squares = 64
¨ Population - Range: 10-0 = 10 - Variance: 64/6 = 10.67 - SD = 3.27 ¨ Sample ¨ Range: 10 ¨ Variance: 64/5 = 12.8 ¨ SD = 3.58 ¨ What is the ONLY
difference between the two formulas? (N vs. n-1)
Standard Deviation
¨ A majority (68% for a normal distribution) of all scores are within one standard deviation on either side of the mean
¨ Only a small minority (5% for a normal distribution) is more than two standard deviations on either side of the mean
Pros and Cons of Standard Deviation
¨ Pros ¤ Used in calculating many other measures. ¤ Average of deviations around the mean. ¤ Majority of data within 1 s.d. above or below the mean. ¤ Combined with mean:
n Efficiently describes a distribution with just two numbers n Allows comparisons between distributions with different scales
¨ Cons
¤ Influenced by extreme scores.