1. Introduction to Statistics
Nyack High School Science Research

2. Researchers often must determine if their data is statistically significant, or a result of a “fluke,” or measurement uncertainties. A researcher will often test a hypothesis on a sample of the population.
sample: a group of people who participate in a study
population: all the people who the study is meant to generalize.

3. Frequency Distribution
A frequency distribution is a table in which all scores are listed, along with the frequency with which each occurs.
Here’s an example of scores from an AP Physics test:
Frequency Distribution rf
Score
frequency
(relative frequency) 56 1
0.077 71 3
0.231 79 80 2
0.154 82 93 95 96 N= 13
1.000

4. Often, data is presented as a frequency distribution of intervals:
Score Interval
frequency rf
56-64 1
0.077
65-73 3
0.231
74-82 5
0.385
83-91
0.000
92-100 4
0.308 N= 13
1.000
The bars in the graph are touching, indicating that the data is continuous.

5. Here’s an example of a frequency distribution for discreet data:
Pet Preference
frequency
dog 6
cat 5
neither 3 N= 14

6. Population mean(µ) = the average of all the scores of the population:
𝑃𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛 𝑚𝑒𝑎𝑛= 𝑠𝑢𝑚 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑐𝑜𝑟𝑒𝑠
OR 𝜇= 𝑋 𝑁
Sample mean ( 𝑿 ): 𝑋 = 𝑋 𝑁
Median = the middle score in a distribution organized from highest-to-lowest, or lowest-to-highest. Referring to the example above, the median score would be ’80.’
Mode = the score with the highest frequency. In the example above, the mode is ’71.’

7. Measures of Variation Range = highest score – lowest score
Standard deviation for a population(σ) is the average distance of all the scores in the distribution from the mean, or central point of the distribution.
𝜎= (𝑋−𝜇) 2 𝑁
Where: X = individual score value

8. What statistical tool do I use?
Determining the relationship between 2 variables
Comparing more than 2 samples
Comparing 2 sample means
ANOVA
(Analysis of Variance)
Correlation/
Regression analysis
T-test
Comparing observed categorical results to expected
Comparing a sample mean to a population mean
Chi-squared
Z-test

9. Standard Scores
z-scores are a measure of how many standard deviation units the individual raw score falls from the mean.
For an individual score, in comparison to a sample:
𝑧= 𝑋− 𝑋 𝑆
For an individual score, in comparison to a population:
𝑧= 𝑋−𝜇 𝜎

10. Standard Scores AP Physics Exam Score (Score – mean) z-score 96 15.46
AP Physics Exam Score
(Score – mean)
z-score 96
15.46
1.32 95
14.46
1.24 56
-24.54
-2.10 71
-9.54
-0.82 93
12.46
1.07 80
-0.54
-0.05 81
0.46
0.04 79
-1.54
-0.13
Mean
80.54
Std. Dev
11.68

11. Null and Alternative Hypotheses
Null hypothesis (Ho): Whatever the research topic, the null hypothesis predicts that there is no difference between the groups being compared. (Which is typically what the research does not expect to find.)
Ex: Say I want to find out if students who attend a review session score higher than those who do not. The null hypothesis would be that the mean score of the group who attended review session would be the same as the mean score of the group who did not attend a review session:
𝜇 𝑟𝑒𝑣𝑖𝑒𝑤 𝑠𝑒𝑠𝑠𝑖𝑜𝑛 = 𝜇 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛
The alternative hypothesis (Ha or H1) would be:
𝜇 𝑟𝑒𝑣𝑖𝑒𝑤 𝑠𝑒𝑠𝑠𝑖𝑜𝑛 > 𝜇 𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑝𝑜𝑝𝑢𝑙𝑎𝑡𝑖𝑜𝑛

12. Null and Alternative Hypotheses
A one-tailed hypothesis is when the direction of difference is predicted. Ex: If I predict that students who attend review session will score higher.
A two-tailed hypothesis is when differences are expected, but the researcher is unsure what they will be. Ex: If I predict that attending a review session will affect scores, but don’t know if the scores would be higher or lower.

13. Null and Alternative Hypotheses
We must determine if our data is “statistically significant.” In other words, we must determine if the data actually supports our hypothesis, or if it just looks that way due to uncontrollable conditions. There are two types of errors:
Type I error: If we reject the null hypothesis, and the null hypothesis is disproved (i.e., the data shows that there is a difference between the population and the sample), but the difference is due to a ‘fluke’ (experimental errors, good guesses, etc.)
Type II error: If we accept the null hypothesis, and the population mean is equal to the sample mean, but this is due to a ‘fluke.’

14. Determining Statistical Significance
We can use either a z-test or a t-test to determine statistical significance. The test we use depends on our data.
z-test: The z-test is used when the population variance is known. It allows the user to compare a sample to a population. The z-test uses the mean and the standard deviation of the sample to determine whether the sample mean is significantly different from the population mean.
t-test: The t-test is used when the population variance is not known. Use the t-test when you have a small sample and you do not know σ.

15. Determining Statistical Significance
Once you have performed a z-test or a t-test, you can plug that value into a program to get your p-value. The discussion of p-values is later.
But wait! There are a other types of t-tests!!! What if you are comparing two different samples, instead of comparing one sample to one population? Then, you must use a different algorithm. We will look at the two possibilities:

16. Determining Statistical Significance
t-test for Independent Groups/Samples
Use this test when you are comparing two samples, representing two populations.
You can compare the two groups in one of two ways:
One group is the control group, and one is the experimental group, or
2. Both groups are experimental, and there is no control.

17. Determining Statistical Significance
t-test for Correlated Groups/Samples
Use this test when you are comparing the performance of participants in two groups, but the same people are used in each group, or different participants are matched between groups (i.e., you are working with pairs of scores for each participant.)
This test is based on the difference score (D), which is the difference between the pairs of scores for each participant.

18. Determining Statistical Significance
t-test for Correlated Groups/Samples
Ex: Eight participants are asked to listen to a single genre of music (hip hop), and then rate the severity of their nightmares on a 1-5 Likert scale (1=mild, 5 = severe.) They are then asked to repeat the process, this time listening to classical music.
Nightmare Severity
Participant
Hip-Hop
Classical
D (difference score) 1 5 4 2 3 6 7 8
Total 10
Mean ( 𝐷 )
1.25

19. Determining Statistical Significance
Chi-Square Tests are nonparametric tests: They do not involve the mean or standard deviation of the population, for one thing.

20. Determining Statistical Significance
1. Chi-Square (χ2) Goodness-of-Fit Test
Used for comparing categorical information (observed frequencies) against what we would expect based on previous knowledge (expected frequencies.)
For example, say a study of students at Nyack High School samples 54 students, and finds 8 of the students (15% of the sample) are overweight or obese. Assume that nationwide, 30% of high school students have been found to be overweight or obese. Observed and Expected frequencies are shown:
Frequencies
Overweight/obese
Not Overweight/obese
Observed (O) 8 46
Expected (E) 16 38

21. Determining Statistical Significance
2. Chi-Square (χ2) Test of Independence
Whereas a χ2 goodness-of-fit test compares how well an observed frequency distribution of one nominal variable fits some expected pattern of frequencies, a χ2 test of independence compares how well two nominal variables fits some expected pattern of frequencies.

22. Determining Statistical Significance
For example, say a study of Nyack High School students looked at whether students who have already taken a health class exercise more than those who have not. We have two variables (taking a health class and exercising.) We find that of the 100 students who have taken a health class, 75 exercise regularly. In the group of students who have not yet taken health, 35 out of 80 exercise regularly. Data is shown in Table 8 below, where the numbers in parenthesis are the expected frequencies, based on the total students polled (180):
Taken Health Class
Yes No
Row Totals (RT)
exercisers
75 (61)
35 (49)
110
Non-exercisers
25 (39)
45 (31) 70
Column Totals (CT)
100 80
180

23. P-values and Statistical Significance
You can now use an on-line p-value calculator to find the p-value. If you’re doing a z-test, simply insert the z-value. If you’re doing a t-test, you need the t-value and the degrees of freedom. For a χ2 test, you need the χ2 value and the degrees of freedom.
An on-line p-value calculator for two-tailed tests is available at:
The p-value for a one-tailed test would be double the p-value for a two-tailed test.

24. P-values and Statistical Significance
So what is a p-value?
A p-value is a probability, with a value ranging from zero to one. A value of zero would mean that, if a random sample of a population were taken, there would be no chance that it would have a larger difference from the total population than what you observed. If the p-value was 0.03, there is a 3% chance of observing a difference as large as you observed.

25. P-values and Statistical Significance
In general, the smaller the p-value, the more “statistically significant” your data is. It’s up to you to set a threshold p-value. Once this is done, every result is either statistically significant or not.
Many scientists refer to data as being either “very significant” if the p-value is below a threshold (usually less than 0.05), and “extremely significant” if below a lower threshold (often less than 0.01).
Sometimes values are flagged with one asterisk for “very significant,” and two asterisks for “extremely significant.”

26. Confidence Intervals
If we don’t know the population mean (µ), we can calculate a confidence interval.
A confidence interval is a range of values which we feel “confident” will contain the population mean, µ.
The confidence level describes the uncertainty involved with a sampling method. A 90% confidence level means that we are 90% confident that the population mean falls within this interval.
Confidence intervals can be calculated from z-scores or t-scores.

27. Confidence Intervals
Referring back to the nightmare study, where the severity of nightmares was rated on a scale of 1-5 (1-mild, 5=severe), the 95% confidence interval in the difference score was calculated to be
Thus, we can say that we are 95% confident that the difference in nightmare severity after listening to classical music is between 0.11 and 2.39 less than the nightmare severity after listening to hip-hop.

28. Correlation Coefficients
When you are looking at a cause-and-effect type of relationship between two variables, a correlation coefficient (r), can be used to measure the strength of the relationship. The value of a correlation coefficient is between 0.00 and 1.00, as follows:
Correlation coefficient (r) Strength of Relationship
± Strong
± Moderate
± None(0.00) to Weak

29. Correlation Coefficients
Sofia.usgs.gov

30. Summary When analyzing data:
Decide what type of experiment you are conducting, and what you are comparing: sample(s) vs. population.
If applicable, plot the frequency distribution.
Look for trends.
Choose a method to test for statistical significance:
z- test
t=test
Chi-Squared test
ANOVA
Regression

31. Claim 1: Money can’t buy you love, but it can buy you a good ball team
Specifically, claim is that baseball teams with bigger salaries win more games than those will smaller salaries
Data are average (mean) salaries and winning percentages for the 2012 baseball season

32. The data TEAM AVG SALARY winning percentage Arizona Diamondbacks
$ 2,653,029
0.5
Atlanta Braves
$ 2,776,998
0.58
Baltimore Orioles
$ 2,807,896
0.574
Boston Red Sox
$ 5,093,724
0.426
Chicago Cubs
$ 3,392,193
0.377
Chicago White Sox
$ 3,876,780
0.525
Cincinnati Reds
$ 2,935,843
0.599
Cleveland Indians
$ 2,704,493
0.42
Colorado Rockies
$ 2,692,054
0.395
Detroit Tigers
$ 4,562,068
0.543
Houston Astros
$ 2,332,730
0.34
Kansas City Royals
$ 2,030,540
0.444
Los Angeles Angels
$ 5,327,074
0.549
Los Angeles Dodgers
$ 3,171,452
0.531
Miami Marlins
$ 4,373,259
Milwaukee Brewers
$ 3,755,920
0.512
Minnesota Twins
$ 3,484,629
0.407
New York Mets
$ 3,457,554
0.457
New York Yankees
$ 6,186,321
0.586
Oakland Athletics
$ 1,845,750
Philadelphia Phillies
$ 5,817,964
Pittsburgh Pirates
$ 2,187,310
0.488
San Diego Padres
$ 1,973,025
0.469
San Francisco Giants
$ 3,920,689
Seattle Mariners
$ 2,927,789
0.463
St. Louis Cardinals
$ 3,939,316
Tampa Bay Rays
$ 2,291,910
0.556
Texas Rangers
$ 4,635,037
Toronto Blue Jays
$ 2,696,042
0.451
Washington Nationals
$ 2,623,746
0.605
The data

33. How is this claim best evaluated? -graph and statistical analysis

34. How is this claim best evaluated? -graph and statistical analysis
Scatter plot

35. How is this claim best evaluated? -graph and statistical analysis
Scatter plot,
Linear regression

36. Conclusion
Money can’t buy you a winning ball team, either

37. Claim 2: Eels control crayfish populations
Specifically, claim is that crayfish population densities are lower in streams where eels are present
Background: dietary studies show that eels eat a lot of crayfish, and old Swedish stories suggest that eels eliminate crayfish
Data are crayfish densities (count along transects, snorkelling) in local streams with and without eels

38. The data River Site Crayfish (no./m^2) eels Croton Green Chimneys
3.225
PEP
0.119
Delaware
Buckingham
0.25 1
Callicoon
Hankins
0.109
Mongaup
Pond Eddy
0.067
Neversink
Bridgeville
0.233
TNC
Shawangunk
Mount Hope
4.53
Ulsterville
1.1
Webatuck
Levin
0.812
Shope
1.719
Still Point
1.4

39. How is this claim best evaluated? -graph and statistical analysis

40. How is this claim best evaluated? -graph and statistical analysis
Bar graph

41. How is this claim best evaluated? -graph and statistical analysis
Bar graph,
t-test
p = 0.02

42. Conclusion
Looks like streams containing eels have fewer crayfish

43. Claim 3: Human life expectancy varies among continents
Data are mean life expectancy for women in different countries

44. The data Africa Asia Americas Europe algeria 75 bangladesh 70.2
argentina
79.9
austria
83.6
cameroon
53.6
china
75.6
brazil
77.4
belgium
82.8
cote d'ivoire
57.7
india
67.6
canada
85.3
bulgaria
77.1
egypt
75.5
indonesia
71.8
chile
82.4
czech rep 81
kenya
59.2
iran
75.3
columbia
77.7
denmark
87.4
morocco
74.9
japan
87.1
mexico
79.6
estonia 80
nigeria
53.4
malaysia
76.9
peru
finland
83.3
south africa
54.1
pakistan
66.9
usa
81.3
france
84.9
zimbabwe
52.7
philippines
72.6
venezuela
germany 83
singapore
83.7
greece
82.6

45. How is this claim best evaluated? -graph and statistical analysis

46. How is this claim best evaluated? -graph and statistical analysis
Bar graph
Note that y-axis doesn’t start at 0

47. How is this claim best evaluated? -graph and statistical analysis
Bar graph,
1-way ANOVA,
p =

48. Anova: Single Factor
SUMMARY
Groups
Count
Sum
Average
Variance
Africa 9
556.1
Asia 10
747.7
74.77
Americas
718.2
79.8
7.7875
Europe
825.7
82.57
ANOVA
Source of Variation SS df MS F
P-value
F crit
Between Groups
2351.6 3
1.42E-07
Within Groups 34
Total 37

49. Conclusion Life expectancy of women appears to differ among continents
(The ANOVA doesn’t tell us which continents are different; further tests would be necessary to test claims about specific continents)

50. Measures of Variation
The standard deviation for a sample (S) formula is similar:
𝑆= (𝑋− 𝑋 ) 2 𝑁
And, when using sample data to estimate the standard deviation of a population – this is called the unbiased estimator of the true population standard deviation (s), use the following formula:
𝑠= (𝑋− 𝑋 ) 2 (𝑁−1)

51. Statistical Distribution Shapes
Fao.org
A: Normal Distribution
B: Positively Skewed Distribution
C: Negatively Skewed Distribution