Statistics

Who Spilled Math All Over My Biology?!

Practical Applications• Population studies:

– Collecting data in the field on a specific population is time consuming and difficult• What is the mean length of le Doge

tails in Beijing? • Much le Doge! Too Many To

Measure!

– A sample group, rather than the whole population, can be examined and the data applied to the larger population

– This is known as a data set• How can we know if the data set

really represents the larger population?– Statistical analysis

The Bell Curve• Data sets of significant size

should show a normal distribution when plotted out – A Bell Curve

• Next, use the data set to calculate:– Standard Deviation– Standard Error– t-Test

• These values can allow one le Doge data set to be applied to other le Doge groups

Standard Deviation• Measures how scattered a data

set is around its mean– Must use a data set with normal

distribution • S= standard deviation• Ʃ= “sum of”• X= value from date set• Ẍ= mean from data set• n= total number of data points• Now we need some le Doge

data

Standard Doge-viation• We get the following data set of le Doge tails (cm):

• First we find the mean (Ẍ) of the data:– Sum of date points/ # of data points– 10.9

• Second, we apply the equation to all the data points

10. 2 11.0 13.5 9.8 11.3 12.3 10.0

8.0 9.9 9.6 9.7 11.6 12.5 11.0

7.9 13.9 12.7 11.5 10.8 11.3 10.4

Standard Doge-viation

• Ẍ = 10.9• Ʃ (x-Ẍ)2 = 49.0• n= 21• S = sq root (49.0/(21-1))

= sq root (2.45)• S= 1.57

X (X-Ẍ)2

10. 2 0.4911.0 0.0113.5 6.769.8 1.21

11.3 0.1612.3 1.9610.0 0.818.0 8.419.9 19.6 1.699.7 1.44

11.6 0.4912.5 2.5611.0 0.017.9 9

13.9 912.7 3.2411.5 0.3610.8 0.0111.3 0.1610.4 0.25

What Does This All Mean?• Mean le Doge tails (Ẍ = 10.9

cm)• 10.9 cm is the height of

our le Doge bell curve• 95% of le Doge tail lengths

fall between the upper and lower limit from the mean (10.9 cm)– Lower limit= Ẍ - (2 x S)– Upper limit= Ẍ + (2 x S)– S= 1.57

• 95% of all le Doge tail in the data set are within: 10.9 ± 3.14 cm

Working the Numbers• Now that we mastered the

date set of one le Doge group, we can apply our findings to rest of the group

• This will save time and energy since we wont need to measure all the le Doge tails of the next group

• The data from group 1 can apply to group 2 as long as they are similar in type

Standard Error (SM)• The estimated standard deviation

of a whole population based on the mean and standard deviation of one date set– Our data set covered le Doge tails of

group A, but we want data on Group B as well

• Because these are normally distributed data sets, we can sure that 95% of the means (Ẍ) of other groups will be ± (2xSM)

• S= standard deviation • n= number of data points

StanDoge Error (SM)• Standard deviation for le Doge tails in

group A; S= 1.57 cm• n= 21• SM= 1.57/4.58 = 0.34 cm• So we can be 95% certain that the

mean le Doge tails in group B is Ẍ (B) = Ẍ (A) ± (2 x SM)– Ẍ (B) = 10.9 ± 0.68 cm

• How would using a sample size of 100 in group A effect our prediction for group B?– Decrease SM range; 10.9 ± 0.26 cm– data is more accurate

Comparing Multiple Data Sets• Using standard error saves time,

however it only works with populations under the same circumstances– Le Doge groups A and B were le Doges

found in Beijing. The data may not apply to le Doges in France.

– The more variables that are not accounted for, the less certain the data becomes

• Paris le Doge tails study was done:– n= 50– Ẍ= 9.5 cm– S= 2.03

• Is there a significant difference between these two groups? How can we tell?

t-Tests• Determines the significance in

differences between means of multiple data sets

• Ẍ1= mean of data set 1

• Ẍ2= mean of data set 2

• S1= standard deviation of set 1

• S2= standard deviation of set 2

• n1= # of data points in set 1

• n2= #of data points in set 2

• t= 3.14• What does this mean?!

Le Doge Tail Lengths (cm)

Beijing Paris

Ẍ 10.9 9.5

S 1.57 2.03

n 21 50

t-Value Table• To understand significant

difference between data points you need 3 things:1) t-Test value2) Degree of freedom from data sets

df= (n1-1) + (n2-1) = 69

3) t-Value Table• Use the df to find the t-value under

0.05– If the t-Test value larger than the t-

value on the chart, you “fail to reject” there is a significant difference between the data sets

– If is it smaller, the two data sets are not significantly different

t-test = 3.14; df= 69T-value= 2.000 3.14 > 2.000 So Pairs le Doge tail lengths and Beijing le Doge tail lengths are significantly different

Time for much practice.

Many homework.

Wow. Such confusion.