✅ Unit 0: Statistics
,Standard deviation:
Equation:
2
Σ(𝑥−𝑥)
s= 𝑛−1
𝑥= average or mean of the data points
n= number of data points (n-1= degrees of freedom)
Σ= the following equation needs to be done for all data points individually, and then add up the
answers
Example:
Data points are 1, 2, 3, 4, 5
Average: 3
(1-3)2=4
(2-3)2=1
(3-3)2=0
(4-3)2=1
(5-3)2=4
Total: 10
n=5, n-1=4
10
s= 4
or 2. 5, which equals 1.58
Application:
When making a graph with a bell curve, standard deviation tells you how much deviation of the
mean is needed in order to encompass most of the data. 1 deviation will get you 65%, 2 will get
you 95%, and 3 will get you 99%. You need standard deviation to tell you how much one
standard deviation is for that specific set of data.
Standard error:
Equation:
𝑠
SE𝑥 (of the mean)=
𝑛
s= standard deviation
n= sample size
Example:
Sample 1: 9, 15
Sample 2: 10.9, 11.9, 12.2, 12.2, 12.9, 12.6, 12.3, 12.3, 12.5, 10.2
Both of these samples have a mean of 12.
Standard error of Sample 1 is 4.24.
4.24
=3
2
,Standard error of Sample 2 is 0.83.
0.83
=0.26
10
Application:
If both of the samples have the same mean, what’s the point of having a bigger sample size?
Standard error of the mean tells you how accurate your information is- the lower the standard
error, the better the data.
Chi squared:
Equation:
2
(𝑜−𝑒)
x2=Σ 𝑒
o=observed
e=expected
x=chi
Example:
A group of 60 people voted whether they liked vanilla, chocolate, or strawberry ice cream better
(here the expected would be 20-20-20). Instead, the numbers were 30, 25, and 5 respectively.
Vanilla: (30-20)2=100/20=5
Chocolate: (25-20)2=25/20=1.25
Strawberry: (5-20)2=225/20=11.25
Total: x2=18.50
Compare to the Chi-Square Table:
Degrees of
Freedom | (p = 0.05) | (p = 0.01)
------------------------------------------
1 | 3.84 | 6.64
2 | 5.99 | 9.21
3 | 7.82 | 11.34
4 | 9.49 | 13.28
5 | 11.07 | 15.09
6 | 12.59 | 16.81
7 | 14.07 | 18.48
8 | 15.51 | 20.09
If the calculated x2 value is larger than or equal to the table value, your results are statistically
significant, so you should reject the null hypothesis. If the calculated x2 is smaller than the table
value, your results are not statistically significant, and you cannot reject the null hypothesis.
,Application:
Chi squared helps test if the data observed differs significantly from the expected value. It’s
good for proving or disprove the null hypothesis.
, ✅ Unit 1: Chemistry of Life
,Standard deviation:
Equation:
2
Σ(𝑥−𝑥)
s= 𝑛−1
𝑥= average or mean of the data points
n= number of data points (n-1= degrees of freedom)
Σ= the following equation needs to be done for all data points individually, and then add up the
answers
Example:
Data points are 1, 2, 3, 4, 5
Average: 3
(1-3)2=4
(2-3)2=1
(3-3)2=0
(4-3)2=1
(5-3)2=4
Total: 10
n=5, n-1=4
10
s= 4
or 2. 5, which equals 1.58
Application:
When making a graph with a bell curve, standard deviation tells you how much deviation of the
mean is needed in order to encompass most of the data. 1 deviation will get you 65%, 2 will get
you 95%, and 3 will get you 99%. You need standard deviation to tell you how much one
standard deviation is for that specific set of data.
Standard error:
Equation:
𝑠
SE𝑥 (of the mean)=
𝑛
s= standard deviation
n= sample size
Example:
Sample 1: 9, 15
Sample 2: 10.9, 11.9, 12.2, 12.2, 12.9, 12.6, 12.3, 12.3, 12.5, 10.2
Both of these samples have a mean of 12.
Standard error of Sample 1 is 4.24.
4.24
=3
2
,Standard error of Sample 2 is 0.83.
0.83
=0.26
10
Application:
If both of the samples have the same mean, what’s the point of having a bigger sample size?
Standard error of the mean tells you how accurate your information is- the lower the standard
error, the better the data.
Chi squared:
Equation:
2
(𝑜−𝑒)
x2=Σ 𝑒
o=observed
e=expected
x=chi
Example:
A group of 60 people voted whether they liked vanilla, chocolate, or strawberry ice cream better
(here the expected would be 20-20-20). Instead, the numbers were 30, 25, and 5 respectively.
Vanilla: (30-20)2=100/20=5
Chocolate: (25-20)2=25/20=1.25
Strawberry: (5-20)2=225/20=11.25
Total: x2=18.50
Compare to the Chi-Square Table:
Degrees of
Freedom | (p = 0.05) | (p = 0.01)
------------------------------------------
1 | 3.84 | 6.64
2 | 5.99 | 9.21
3 | 7.82 | 11.34
4 | 9.49 | 13.28
5 | 11.07 | 15.09
6 | 12.59 | 16.81
7 | 14.07 | 18.48
8 | 15.51 | 20.09
If the calculated x2 value is larger than or equal to the table value, your results are statistically
significant, so you should reject the null hypothesis. If the calculated x2 is smaller than the table
value, your results are not statistically significant, and you cannot reject the null hypothesis.
,Application:
Chi squared helps test if the data observed differs significantly from the expected value. It’s
good for proving or disprove the null hypothesis.
, ✅ Unit 1: Chemistry of Life
