Introduction to Quantitative

Skills/Graphing August 19, 2014

Objectives• Mean, median, mode, range• The nature of science • Observations vs. Inferences• Populations vs. Samples• Null vs. Alternative Hypotheses• Designing an Experiment • Graphing• Basic Statistics

Students need to be able to determine what type of graph (e.g., histogram, line graph) most appropriately reflects their collected data and then create the graph and use it to draw conclusions, make predictions, and pose questions for further investigation.

Review• Mean: average of a set of numbers• Median: middle number if you add them up smallest to largest• Mode: most frequent number• Range: largest number minus smallest number

Mean, Median, Mode, Range Example 13, 18, 13, 14, 13, 16, 14, 21, 13

Mean Median Mode Range

(13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15

13, 13, 13, 13, 14, 14, 16, 18, 21

(If even number, take average of middle two numbers)

13 21-13 = 8

Nature of Science• Science is about finding answers to problems

through investigations• Science is about questioning everything (skepticism).

Don’t tell me something is true, prove it to me!• Science is about only accepting something as the

“right answer” if enough evidence has been built to support it and it can be replicated by others

• There is no emotions in science….but…• Creativity is a vital, yet personal, ingredient in the

production of scientific knowledge.

The Science Community • The science community questions

findings, replicates hypotheses, and peer reviews publications

• By it’s very nature, the science community is a scientists biggest critic

• Real debates should be between two parties who both hold massive amounts of evidence

Science is Dynamic Scientific knowledge is simultaneously reliable and tentative. Having confidence in scientific knowledge is reasonable while realizing that such knowledge may be abandoned or modified in light of new evidence or reconceptualization of prior evidence and knowledge

Examples of Science Gone Wrong• Is the earth the center of the universe? Is the earth flat?• Newton’s three laws of motion – considered fact until we got into

Einstein's Theory of Relativity• Even NASA was wrong!

Carbon is the element of Life for a reason

The Nature of Science• It’s also just as bad to ignore mounting evidence

against virtually undisputed scientific theories• Evolution• Global Warming

• The nature of science is to QUESTION, CHALLENGE, BE OBJECTIVE, and THINK CRITICALLY

• It’s OK to change your mind!

Basis of the Scientific Method • It’s okay if your hypothesis is wrong!!!!

• Your experimental design is critical!

• ALL scientists must be FLUENT in statistics

• If your data supports your hypothesis, does that mean your conclusion is fact? Hmmm

• There are NO facts in science. Whhhaaa?

Observation vs. Inference • Observation: using your five senses to

observe descriptive things• Qualitative• Quantitative

• Inference: taking all of the observations you have made and trying to explain them

• You do this every day, whether you are meaning to or not

Observations lead to questions• We make observations everyday with our senses

• Qualitative: descriptive (The cat is grey)• Quantitative: numerical (The cat weighs 22 lbs)

• A gathering of sample observations help us make inferences about populations with our reasoning skills

• We can never know anything for sure about the true population – it is often too large and ever changing

Evidence• Scientists must generate data as evidence • Data can be:

• Quantitative: numbers, data measured, uses instruments• Qualitative: descriptive, data observed, uses senses

Quantitative Data • Can be either: Continuous of Discrete

Sample vs. Population

Quantitative data you collect will fall into 1 of 3 categories• Parametric data

• Normal curve = bell curve = parametric data• Describes most populations • usually decimals included (continuous)• The less bias your sample is, the closer to the true mean it will be

• Nonparametric data • not normally distributed (bell-shaped)• includes outliers• Qualitative (small, medium, large)

• Frequency/count data• Ex. How many flies with certain type of wing• This is a way to make qualitative data quantitative data

Parametric or Nonparametric?

Non-Parametric Data • Generally, the parameters calculated for

nonparametric statistics include medians, modes, and quartiles, and the graphs are often box-and-whisker plots

• Why would we use medians instead of means for non-parametric data?

Hypothesis

Formulating Hypotheses

• Null hypothesis (Ho): No observed effect; the opposite of what you’re testing• Alternative hypothesis (HA) or (H1): An effect was observed; the claim your

testing • We either fail to reject the null () or reject the null ()• To begin with we always assume the null is true (like innocent until proven

guilty)• As a scientist, the alternative hypothesis is our friend and being able to

reject the null is very exciting because if a significant effect was observed then we get to publish our results!

Possible Hypothesis Outcomes

Examplein #’s

• (Ho): μ < 2.7 • (HA): μ > 2.7

• (Ho): • (HA):

Examplein words

Example

hypothesis that children have a higher IQ if they eat oily fish for a period of time.

Designing an Experiment• Variables

• Independent: what we control/change• Dependent: what changes in response to independent

• Constant Variable – variables that remain unchanged so we can accurately test out dependent variable

• Control Group• Sample of our population (N) must be LARGE &

RANDOM• Repeated Measures

“___(blank)___ depends on ___(blank)___”

Identify the experimental parameters

• Independent Variable: • Dependent Variable: • Control Variable: • Control Group:

Example

hypothesis that children have a higher IQ if they eat oily fish for a period of time.

Tables and Graphs• A graph is a visual representation of a data set• On your AP test you will be given a data table and must decide which

type of graph will be used• What’s the difference between a bar graph and a histogram?

Histogram

Bar Graphs• Use two compare two samples of categorical or count data. You

will need to compare the calculated means with error bars. Example: qualitative: eye color.

Always use Sample Standard error bar. (Sample error of the mean) (SEM)

Bar graph with only the mean plotted.

Scatterplots• Used to explore associations between two variables visually. One

variable is measured against another. It is looking for trends or associations. Plotting of individual data points on an x-y plot.

Scatterplots

Bell-curved

Sine wave-like

Box-and-whisker plot• Allows comparison of two samples of nonparametric data

(data that does not fit a normal distribution)• Medians and quartiles

Histogram - nonparametric• For frequency data • Can turn numbers into categories!

Cheat sheet of what type of graph to use when

• Pie charts are you to represent percentages of categorical data mostly, but they have a major downfall: they cannot present the error

• Only if you are graphing rate (something changing over time), than you will do a line graph

• Rate equals “rise over run” or the slope of the graph

• If you have frequencies of a number range histogram• Use bar graph when one variable is categorical, and one is numerical• Use scatterplot when both variables are numerical and you want to

see a trend (best fit line)

Don’t lose points!• Be neat!• Don’t forget a TITLE: usually same as

the table title it is representing• LABEL AXES: dependent variable goes

on y-axis; independent goes on the x-axis

• Numbers should ascend from bottom to top or from left to right

• Numbers of axes must be EVENLY spaced

• Don’t necessary have to start data at origin…explanation?

Crappy Graph Game

Which one represents the data right? Why?

This graph is wrong on many levels…

And this one…

This one is a bit harder…

Interpreting General Curves of Graphs

• Bell-shaped curve: associated with random samples and normal distributions.

• Concave upward curve: associated with exponentially increasing functions (for example, in the early stages of bacterial growth) and then plateauing upon saturation/carrying-capacity

• A sine wave–like curve: is associated with a biological rhythm

The Importance of Standard Error

McDonalds Wendy's0

5

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35

Blah, blah, blah, blah

• Ho: There is no difference between McDonalds & Wendy’s• HA: There is a difference between McDonalds & Wendy’s

McDonalds Wendy's0

5

10

15

20

25

30

35

40

45

blah blah blah

End of first half of material

Statistics is the Basis of Science!• Statistics is the MOST important math in Ms. Grapes’ opinion • If you are going into a science-based career, take it!• Even if you are not, knowing statistics will help you think

quantitatively, AKA make better life decisions and think for yourself.

Determining Significance• How can we tell if we accept or reject

Ho? AKA How do we know when the differences we see in data are significant?

• In simple statistics we need two things• The average • The standard deviation (σ or s)

• You can be accurate without being precise

Empirical Rule

• The Empirical Rule/67-95-99.7 Rule/three sigma rule describes the bell curve

• 95% confidence interval: a measure of the reliability of an estimate

Standard Error vs. Standard Deviation • Standard error allows for inference of how sample mean matches

up to the true population mean.• A distribution of the sample means helps define boundaries of

confidence in our sample. 1 SE describes 67% confidence range. 2 SE defines 95% certaintyConfidence limits contains the true population mean.95% confident: larger range to contain the population mean (95% of the data found here)67% confident: narrower range to contain the population mean. SE – inference in which to draw conclusionsSD – just looking at data95% is trade off for never being 100% sure.

From APA sample mean of ±1 SE describes the range of values about which an investigatorcan have approximately 67% confidence that the range includes the true populationmean. Even better, a sample with a ±2 SE defines a range of values with approximatelya 95% certainty. In other words, if the sampling were repeated 20 times with the samesample size each time, the confidence limits, defined by ±2 SE, would include the truepopulation mean approximately 19 times on average. This is the inference; it is a statisticthat allows investigators to gauge just how good their estimate of the true populationmean actually is.

Best-fit line, linear regression, and r

• A best fit line indicates the general trend• A regression line is a best fit line that summarizes all

the points into a single linear equation…for scatterplot• Positive correlation/direct relationship: both variables

go up together• Negative correlation/Indirect relationship: as one

variable goes up, the other gown down. • r-value (from -1 to 1)

• provides an estimate of the degree of correlation. • 0 equals no correlation• 1 is a perfect positive correlation and -1 is visa versa

Standard Deviation (σ)• The average value away from the average• The smaller σ, the better• If your results were ‘precise’ you will have a small std. dev.• The standard deviation is the square root of the variance

http://www.mathsisfun.com/data/standard-deviation.html

Review so far with an example• PROBLEM: Lays mini bags of potato chips should weigh 20 oz. but

they feel heavier to me.• HYPOTHESIS:

• Ho: μ = 20 oz. • HA: μ > 20 oz.

• EXPERIEMENT & COLLECT DATA:• n=36 x = 22 oz. σ = 4 oz. α = 0.05How do we know if this is significant or not?

• ANALYZE DATA: find P-value

Statistics• The data analysis tools used, such as SE, mean, variance, median, etc.

are now used to calculate comparative test statistics.

• These tests determine a probability and distribution of the samples – the null distribution.

• A critical value of the probability of whether the results or even more extreme results occur by chance alone, if the null hypothesis is true (probability value, or p-value of less than 5%)

Statistical Tests…ugh

The P-Value• The P-Value (P): the probability of obtaining a test statistic result at least

as extreme as the one that was actually observed, assuming that the null hypothesis is true.

• P-values are coupled with significance (alpha) levels• If P < 0.05 than we reject the null • AKA, if the null were true, than we would see our result 5% of the time….uhhh

that’s barely anytime at all…so we reject the null.

How to find P-value• First must find the z-statistic• Now we need a z-table

• Alpha =

n=36 x = 22 oz. σ = 4 oz. α = 0.05

(22-20) = 3.00 (4/√36)

How to find P-value• First must find the z-statistic• Now we need a z-table

• Alpha = 1- .9989 = 0.0011

• .0011 is less than .05 so we can reject the null hypothesis and accept the alternative hypothesis

• At the 0.05 significance level the data provide significant evidence to conclude that the mean weight of lays mini bags of chips differs from 20oz. We are 95% confident that the mean weight of mini bags of chips is greater than 20 ounces.

μ =20 n=36 x = 22 oz. σ = 4 oz. α = 0.05

(22-20) = 3.00 (4/√36)

My caveat…• There are t-statistics and z-statistics

• T distributions are for small sample sized (n < 30) • Z distributions are for large samples (n>30)

• There are one-tailed and two-tailed tests• If alternative hypothesis has a “not equal to” than two-tailed• If alternative hypothesis has a “greater than” or “less than” than one-tailed

• There are one-tailed and two-tailed tests• Student’s paired t-test, Chi-square, ANOVA, linear regression…trust

me, take a statistics class before you graduate

Chi-squared Test • Very popular in science…will be on your AP test• Use it for frequencies of more than one outcome• Makes comparison between the data collected and what we expect. • We will use this test most often. Genetics, animal behavior. • Basic idea is testing your chi-squared value against a “critical value” (if

your value is less than critical value than fail to reject the null)• Ho: There is no significant difference between the observed and

expected frequencies

Chi-Squared Example• I flip a coin and get heads 28 times and tails 22 times• The expected value was 25/25• Need to find critical value

• Degrees of freedom (df) is (n-1)• Since we have two outcomes (heads & tails) our df is 1• Critical value is 3.84

• Now find our chi-square values for each observation

Always use 0.05

.72 is less than our critical value of 3.84, therefore, We accept the null hypothesis

AnotherExample

The suggested path:• 1. Choose a statistical test based on the question and types of data. • 2. State the null and alternative hypotheses • 3. Design and carry out the investigation.• 4. Conduct a data analysis, and present graphical and tabular

summaries of the data. Need to decide if data is normal or not!• 5. Carry out the statistical tests. Include the sample size, test statistic

chosen, and p-values in the data reports.• 6. Make a conclusion, always stating the amount of evidence in terms

of the alternative hypothesis.

Quantitative/Statistics Unit • Mean/median/mode/range• Observations vs. Inferences• Writing null and alternative hypotheses• Designing a valid experiment (variables, control groups, repeated

measures)• Bell curve (normal curve)• Graphing your results from a table (with error bars)• Statistics (p-value, chi-square)

Grid-in Questions on AP Exam

Before we end, let’s do one together

What better techniques could she have used? Is this data continuous or discrete?How would you graph this?How could you find out if the data is normally distributed?

Is this curve normal or not?

We are dealing with a population and so most likely, it would be normal. How could she have made it more parametric? What’s next?

References• www.randi.org

• www.badscience.net

• MythBusters

• Ted Talks “Battling Bad Science”• http://www.ted.com/talks/ben_goldacre_battling_bad_science

• https://www.khanacademy.org/math/probability/statistics-inferential/confidence-intervals/v/confidence-interval-1

Handbook of Biological StatisticsGood Resource

• http://udel.edu/~mcdonald/statintro.html

• I find that a systematic, step-by-step approach is the best way to analyze biological data. The statistical analysis of a biological experiment may be broken down into the following steps:

• Specify the biological question to be answered. • Put the question in the form of a biological null hypothesis and alternate hypothesis. • Put the question in the form of a statistical null hypothesis and alternate hypothesis. • Determine which variables are relevant to the question. • Determine what kind of variable each one is. • Design an experiment that controls or randomizes the confounding variables. • Based on the number of variables, the kind of variables, the expected fit to the parametric assumptions, and the

hypothesis to be tested, choose the best statistical test to use. • If possible, do a power analysis to determine a good sample size for the experiment. • Do the experiment. • Examine the data to see if it meets the assumptions of the statistical test you chose (normality, homoscedasticity,

etc.). If it doesn't, choose a more appropriate test. • Apply the chosen statistical test, and interpret the result. • Communicate your results effectively, usually with a graph or table.

• http://www.radford.edu/jkell/statsgraphs.pdf• http://www.biologyforlife.com/graphing.html