Statistical AnalysisHL Biology

2010

1.1.1: State that error bars are a graphical representation 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 values fall within on standard deviation of the mean

1.1.4: Explain how the standard deviation is useful for comparing the means and spread of 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 the appropriate tables

1.1.6: Explain that the existence of a correlation does not establish that there is a casual relationship between variables

Topic 1.1: Statistics

Steps to Scientific Method◦ Ask a Question/Observation ◦ Do Background Research ◦ Construct a Hypothesis ◦ Test Your Hypothesis by Doing an

Experiment ◦ Analyze Your Data and Draw a Conclusion ◦ Communicate Your Results

Why do we use Stats in Biology?

In science, observations result in Numbers!!◦ What is the height of bean plants growing in

sunlight compared to the height of bean plants growing bean plants growing in shade? Write Specific Question!!! Set up Experiment Sample!!

Why do we use Stats in Biology?

Statistics is a branch of mathematics which allows us to sample small portions from habitats, communities, or biological populations, and draw conclusions about the larger populations◦ Measures the differences and relationships

between sets of data Using example from previous page—how would be

determine if there is a difference between the control and variable?

Why do we use Stats in Biology?

Mean: is the average of data points Range: range is the measure of the spread

of data Standard Deviation: is a measure of how

the individual observation of data set are dispersed or spread out around the mean

Statistical analysis

1. Sandra is playing in a tennis doubles tournament. The rules say that the average age of the pair of players on each side must be ten years old or younger. Sandra is eight years old. Her partner must be _____ years old or younger.

2. Juan has played in four baseball games this season. He struck out an average of twice per game. In the last three games, he didn’t strike out at all. How many times did he strike out in the first game of the season? 

3. Jerome took five spelling tests in the last marking period. He scored 100% in all but one. His lowest score was 80%. What was his mean score for the spelling tests in the last marking period?

4. Lucy bought seven pens. Four of the pens cost a dollar each. Three of the pens cost 30 cents each. What was the average cost of each pen?

Mean Problems

Cheryl took 7 math tests in one marking period. What is the range of her test scores?   89,  73,  84,  91,  87,  77,  94

The Jaeger family drove through 6 midwestern states on their summer vacation. Gasoline prices varied from state to state. What is the range of gasoline prices?$1.79,  $1.61,  $1.96,  $2.09,  $1.84,  $1.75

A marathon race was completed by 5 participants. What is the range of times given in hours below?  

2.7 hr,  8.3 hr,  3.5 hr,  5.1 hr,  4.9 hr

Range Problems

Are a graphical representation of the variability of data◦ Can be used to show either the range of data or

the standard deviation of a graph◦ The value of the standard deviation above the

mean is shown extending above the top bar of the histogram and the same standard deviation below the top of each bar of the histogram

Error Bars

We will use standard deviation to summarize the spread of values around the mean and to compare the means and spread of data between two or more sample◦ In a normal distribution, about 68% of all values

lie within ±1 standard deviation of the mean◦ This rises to about 95% for ±2 standard deviation

from the mean

Standard Deviation

Use Bean plant experiment as an example!

If we plot all the data points of the 100 plants we should get a bell shaped curve

Y-axis- number of bean plants X-axis-heights

Standard Deviation

Most sets of numbers are not this perfect!!!◦ Sometimes the bell-shape is very flat

This indicates that the data is spread out widely from the mean

◦ Sometimes the bell shape is very tall and narrow This indicates the data is very close to the mean and

not spread out

Standard Deviation

The standard deviation tells us how tightly the data points are clustered together◦ When standard deviation is small—data points are

clustered very close◦ When standard deviation is large—data points are

spread out

Standard Deviation

Why is this important?◦ Tell how many extremes are in the data

Standard Deviation

Remember in statistics we make references about a whole population based on just a sample of the population

Standard DeviationHeight of bean plants in the sunlight in centimeters ±0.1 cm

Height of bean plants in the shade in centimeters ±0.1 cm

124 131

120 60

153 160

98 212

123 117

142 65

156 155

128 160

139 145

117 95

Total: 1300 Total: 1300

First, determine the mean for each set of data

Second, determine the range for each set of data

Look at the two sets of data is there variation among the two sets of data

Standard DeviationHeight of bean plants in the sunlight in centimeters ±0.1 cm

Hei128ght of bean plants in the s1hade in centimeters ±0.1 cm

124 131

120 60

153 160

98 212

123 117

142 65

156 155

128 160

139 145

117 95

Total: 1300 Total: 1300

How can we mathematically quantify the variation that we observed?◦ Standard Deviation

Use Graphing Calculator: Code: 4242P www.heinemann.co.uk/hotlinks

Use website : http://easycalculation.com/statistics/standard-deviation.php

Use excel: http://www.gifted.uconn.edu/siegle/research/Normal/

stdexcel.htm

Standard Deviation

What does this number mean?◦ High Standard deviation—indicates a very wide

spread of data around the mean. This would make us question the experimental

design Experimental error

Standard Deviation

Residents of upstate New York are accustomed to large amounts of snow with snowfalls often exceeding 6 inches in one day.  In one city, such snowfalls were recorded for two seasons and are as follows (in inches):

8.6, 9.5, 14.1, 11.5, 7.0, 8.4, 9.0, 6.7, 21.5, 7.7, 6.8, 6.1, 8.5, 14.4, 6.1, 8.0, 9.2, 7.1

What are the mean and the population standard deviation for this data, to the nearest hundredth?

Standard Deviation Problem

Neesha's scores in Chemistry this semester were rather inconsistent:  100, 85, 55, 95, 75, 100.  For this population, how many scores are within one standard deviation of the mean?

Standard Deviation Problem

The number of children of each of the first 41 United States presidents is given in the accompanying table.  For this population, determine the mean and the standard deviation to the nearest tenth. How many of these presidents fall within one standard deviation of the mean?

Standard Deviation Problem

All of these (mean, range, and standard deviation) still does not tell us whether there is a difference between two sets of data

The t-test compares two sets of data and tells us if there is a significant difference between the two sets of data

Look at Table of t values on Pg 7 in handout

T-test

p(probability value)-chance alone could produce the difference◦ 0.5, 0.2, 0.1, 0.05, 0.01, 0.001

Most science uses 0.05 column.◦ This means there is a 5% chance that the

difference between the two numbers are by chance alone

◦ 95% chance that the difference between the two numbers are because of the variable in your experiment. (bean plant-variable would be light)

T-test

Degrees of Freedom◦ This is the sum of the sample sizes of each of the

two groups minus two. T-test:

http://www.graphpad.com/quickcalcs/ttest1.cfm

www.Heinemann.co.uk/hotlinks

T-test

Now you have the p-value, the t-test value, and the df. What does all these numbers mean?◦ You are going to compare the p-value with your t-

test results◦ If:

T-test value is greater than P-value—significant T-test value is smaller than P-value—not significant

Significance

A research study was conducted to examine the differences between older and younger adults on perceived life satisfaction. A pilot study was conducted to examine this hypothesis. Ten older adults (over the age of 70) and ten younger adults (between 20 and 30) were give a life satisfaction test (known to have high reliability and validity). Scores on the measure range from 0 to 60 with high scores indicative of high life satisfaction; low scores indicative of low life satisfaction. The data are presented below. Compute the appropriate t-test.

Practice Problems t-testOlder Adults Younger Adults

45 34

38 22

52 15

48 27

25 37

39 41

51 24

46 19

55 26

46 36

Mean = Mean =

S = S =

S2 = S2 =

A researcher hypothesizes that electrical stimulation of the lateral habenula will result in a decrease in food intake (in this case, chocolate chips) in rats. Rats undergo stereotaxic surgery and an electrode is implanted in the right lateral habenula. Following a ten day recovery period, rats (kept at 80 percent body weight) are tested for the number of chocolate chips consumed during a 10 minute period of time both with and without electrical stimulation. The testing conditions are counter balanced. Compute the appropriate t-test for the data provided below.

Practice Problems t-test

Stimulation No Stimulation

12 8

7 7

3 4

11 14

8 6

5 7

14 12

7 5

9 5

10 8

Mean = Mean =

S = S =

S2 = S2 =

Researchers want to examine the effect of perceived control on health complaints of geriatric patients in a long-term care facility. Thirty patients are randomly selected to participate in the study. Half are given a plant to care for and half are given a plant but the care is conducted by the staff. Number of health complaints are recorded for each patient over the following seven days. Compute the appropriate t-test for the data provided below.

Practice Problems t-testControl over Plant

No Control over Plant

23 35

12 21

6 26

15 24

18 17

5 23

21 37

18 22

34 16

10 38

23 23

14 41

19 27

23 24

8 32

Mean = Mean =

S = S =

S2 = S2 =

The correlation is one of the most common and most useful statistics. A correlation is a single number that describes the degree of relationship between two variables.

Correlation does not mean causation

We use the symbol r to stand for the correlation.

r will always be between -1.0 and +1.0.◦ if the correlation is

negative, we have a negative relationship

◦ if it's positive, the relationship is positive.

Correlation does not mean causation

Correlation is not a cause◦ To find the cause of the observed correlation

requires experimental evidence◦ Assess the statements (handout) to determine

whether you think they represent a correlation or if there is a causal connection

Correlation does not mean causation

Practice Problems