Statistical Analysis

Topic – 1.1.1-1.1.6

Math skills requirements

Syllabus Statements• 1.1.1: State that error bars are graphical

representations of the variability of data• 1.1.2: calculate the mean and standard deviation of a

set of values• 1.1.3: State that the term standard deviation is used to

summarize the spread of values around the mean and that 68% of the values fall within one standard deviation of the mean

• 1.1.4: Explain how the standard deviation is useful for comparing the means and the spread of the data between two or more samples

• 1.1.5: deduce the significance of the difference between two sets of data using calculated values for t and appropriate tables

• 1.1.6: Explain that the existence of a correlation does not establish a causal relationship between two variables.

Error Bars

• Bars on a graph only show means and can be misleading

• Error bars show variability around the mean

• Can be used to show range, standard deviation or standard error

Means can look different

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But really not be

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Given a set of data can you calculate mean and stdev?

• In calculator • Stat key• Edit and enter your

list(s)• Stat key again • Calc and 1-var stats

then specify your list• Which one is the

mean?• Which one is the

standard deviation?

• Use the following data as an example

• 170, 160, 150, 175, 180, 175, 190, 165

• The mean is 170.63

• The standard deviation is 12.37 (use the s value)

So what is the Standard Deviation?

Standard Deviation is just

• A numerical measure of the spread of the data around the mean

• The absolute number doesn’t mean a lot• Look at the number in relation to the mean• If you mean is 100 and your standard deviation

is 1 then its tiny• If your mean is 1.5 and your standard deviation

is 1 then that is pretty significant• Rule of thumb is if Sx/mean > .20 then its getting

up there

So by definition the Standard deviation marks off discrete intervals under a bell curve

• In a normal distribution (bell curve) remember the 68, 95, 99.7 RULE

• 68% of observations are within 1 stdev of the mean, 95% within 2 stdev, 99.7% within 3 stdev

• Mean of 18 stdev of 4.5 => 68% = 13.5-22.5 • Now can compare mean and spread of 2

distributionssmall stdev = values tightly cluster

around the mean (little variability)

large stdev = values spread out around the mean (large variability)

Using Standard Deviation to compare Variability around means

Step 8: Does your data really show an effect?

• Statistics give power to your results

• Is your result just chance or is it caused by your Independent Variable (IV)?

• Statistics uses probability to determine how likely it is that your results are just random

• You should understand T-test, linear regression analysis

Statistics: T-test

• Compares the means of two populations which are normally distributed, with sample size of at least 10.

• A way to tell if means of two groups are actually different from each other. (Or conversely looks at the amount of overlap between the two)

• Accounts for the mean and variability of the data

• Two tailed unpaired T-test is expected

• Not expected to calculate T

• So while we usually say that if the p value is < .05 then there is a significant difference

• They want you to go from a T table as follows

t table with right tail probabilities

                     

df\p

0.40 0.25 0.10 0.05 0.025 0.01 0.005 0.0005

1 0.3249 1.0000 3.0776 6.3137 12.706 31.820 63.656 636.61

2 0.2886 0.8164 1.885 2.9199 4.3026 6.9645 9.9248 31.599

3 0.2766 0.7648 1.637 2.3533 3.1824 4.5407 5.8409 12.924

4 0.2707 0.7406 1.5332 2.1318 2.7764 3.7469 4.6040 8.6103

5 0.2671 0.7266 1.4758 2.0150 2.5705 3.3649 4.0321 6.8688

To calculate the df you take the total number of samples and subtract 2T value must exceed the value in a given cell to be that p valueThink of those p values as percentages look at p = .05 column

So back to our graphs

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•Is there an actual difference between the means?•Conduct T-test if p < 0.05 then there is an actualdifference, otherwise its just a chance event

• So mean boys height was 71

• And mean girls height was 64

• And the T value was t = 0.082

• And the T critical value in the table for p=.05 was t=2.002using appropriate degrees of freedom

• So our t was too small meaning that the means are NOT significantly different

Statistics: Linear Regression

• Is there a relationship between two variables that are measured in an experiment?

• Works with scatterplots with a line of best fit

e.g. Height and weight data, age and weight data

• Does change in one variable predict change in another?

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•Does change in Length predict a change in Weight?•Is there a positive or negative correlation (slope)•r = correlation coefficient – measures the strength of the linear association between 2 quantitative variables

Correlation

• df = number of points in scatterplot – 2 (x and y axis)

• Calculate r with equation or program

• Use table to determine the critical value for the number of points you are using

• r must exceed that number for a significant relationship (correlation) to be present

But remember

• The existence of a correlation does not indicate causation

• So if people with bigger hands have bigger feet that does not mean that a change in hand size causes a change in foot size

• Rather they are both caused by something else…