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Statistics Or Do our Data mean Diddly?

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Page 1: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Statistics

OrDo our Data mean Diddly?

Page 2: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Why are stat important

• Sometimes two data sets look different, but aren’t

• Other times, two data sets don’t look that different, but are.

Page 3: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Normal Distribution

• The data are centered around the mean

• The data are distributed symmetrically around the mean

http://en.wikipedia.org/wiki/File:Planche_de_Galton.jpg

Page 4: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Mean μ vs

• The entire population mean is μ• Sample population mean is • As your sample population gets larger,

• Data Set– 2, 3, 4, 4, 5, 5, 6, 6, 7, 7, 8, 9

• Mean

x

=2 + 3+ 4 + 4 + 5 + 5 + 6 + 6 + 7 + 7 + 8 + 9

12

x

x

x → μ

Series10

0.51

1.52

2.53

Page 5: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Standard Deviation

• Describes how data are expected to vary from the mean

• σ is s.d. of population s is s.d. of sample

http://en.wikipedia.org/wiki/File:Standard_deviation_illustration.gif

• μ = 50• σ = 20

s =1

n −1 i=1

n

Σ x i − x ( )2

σ =1

N i=1

N

Σ x i −μ( )2

Page 6: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Meaning of Standard Deviation

• Red, Green, Blue all same mean

• Different standard deviation

Page 7: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Meaning of Standard Deviation

• Data with a larger spread (blue and green) have a larger Standard Deviation

Page 8: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Standard Deviation

• 68% of values are within 1 standard deviation• 95% of values are within 2 standard deviations

of the mean

Page 9: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

T-tests

• Assumptions– Data are normally distributed– The mean is independent of the standard deviation

• μ ≠ f(σ)

• Various types– One sample t-test

• Are these data different than the entire population?

– Two sample t-test• Do these two data sets come from different populations?

– Paired t-test• Do individual changes show an overall change?

Page 10: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Use t-test to compare means

• We have and – Do they come from different populations?

• Are and different?

• Null Hypothesis Ho: – =

• Alternative Hypothesis Ha:– >

• t statistic tests Ho. If t < 0.05, then reject Ho and accept Ha

x 1

x 2

μ1

μ2

x 1

x 2

x 1

x 2

Page 11: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Application

t =x 1 − x 2

s

n1n2n1 + n2

s =

(x i − x 1)2

set1

∑ + (x j − x 2)2

set2

∑n1 + n2 −2

s =1.2

Hawks Cyclones9 48 67 56 27 48 5

X1 7.5 X2 4.3s1 1.0 s2 1.4

t =7.5 − 4.3

1.2

6 ×6

6 + 6t = 4.6

tcalc = 4.6t95 = 2.2tcalc > t95

HAW

KS W

IN!

Go to table in notes to find t95 with 11 degrees of freedom (12-1)

Ho : x 1 = x 2HA : x 1⟩x 2

(The excel sheet does a different comparison)

Page 12: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Exercise 99%

t =x 1 − x 2

s

n1n2n1 + n2

s =

(x i − x 1)2

set1

∑ + (x j − x 2)2

set2

∑n1 + n2 −2

s = ?

MIT Harvard100 46

87 5456 7687 9298 8790 60

X1 86.3 X2 69.2s1 15.9 s2 18.6

tcalc = 1.79t99 = ?tcalc ? T99

Go to table in notes to find t99 with 11 degrees of freedom (12-1)

Ho : x 1 = x 2HA : x 1⟩x 2

(The excel sheet does a different comparison)

t=?

Page 13: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Confidence Intervals

• Use t to find interval containing μ if is known

• Example:t95 = 2.6

6.4 < μ <8.6I am 95% confident that the population mean lies between 6.4 and 8.6

x

μ =x ±ts

nHawks Cyclones

9 48 67 56 27 48 5

X1 7.5 X2 4.3s1 1.0 s2 1.4

μ1 = 7.5 ±2.6 ×1.0

6

Page 14: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Exercise

• You find confidence interval of μ1 at the 99% confidence interval

• t99 = ?

μ =x ±ts

n

Hawks Cyclones9 48 67 56 27 48 5

X1 7.5 X2 4.3s1 1.0 s2 1.4

4.032

5.85 < μ < 9.15

?< μ < ?

Page 15: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Figure 2

Page 16: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look
Page 17: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look
Page 18: Statistics Or Do our Data mean Diddly?. Why are stat important Sometimes two data sets look different, but aren’t Other times, two data sets don’t look

Targeted cells showed green fluorescence via flow cytometry at expected frequency.

Jonnalagadda, et al. 2005 DNA Repair. (4) 594-605.