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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.