Assessment Statements• 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 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 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 labels

• 1.1.6 – Explain that the existence of a correlation does not establish that there is a causal relationship between two variables

STATISTICS!!!

The science of data

http://www.worldometers.info/

Scientific Method - Review of Experimental Terminology and Concepts

• Independent Variable: The variable that is manipulated in an experiment. For example, in an experiment to test the affect of light intensity on plant growth, the light intensity would be the independent variable because you are manipulating (changing) the intensity to see what affect it has on plant growth.

• Dependent Variable: This is the variable that is measured in

an experiment. It is the variable that you are measuring to see what affect the independent variable has on it. In the example above, the Dependent Variable is Plant Growth

Scientific Method - Review of Experimental Terminology and Concepts

• Experimental Group(s): All experiments have Experimental Groups. These are the organisms (in this case, individual plants) that are exposed to the same light intensity. For example, 10 plants exposed to 20 foot-candles of light intensity would constitute one experimental group. Ten different plants exposed to 40 foot-candles of light would be another experimental group and so on.

• • Control Group: Some experiments have a Control Group. If there is a

Control Group, it is the group of organisms (in this case plants) that are not exposed to the factor being tested. In our example, the Control Group could perhaps be a group of plants in total darkness (ie. the absence of the factor- light intensity – being tested). This group always serves as a standard of COMPARISON for the Experimental Groups. Think of another experiment that would have a control group and identify the control group below.

Scientific Method - Review of Experimental Terminology and Concepts

The Affect of _______________________ on plant growth

The control group in this experiment would be __________________________________________________________________________

Scientific Method - Review of Experimental Terminology and Concepts

• Controlled Variables: Even though some experiments don’t have a Control Group, all good experiments have Controlled Variables. These are the variables that could affect the Dependent Variable (in this example plant growth) other than the Independent Variables. In our plant example, controlled variables would include

– Amount of water the plants receive– Temperature that the plants are exposed to– Nutrients in the soil

These variables can be controlled in two ways.• Physical Control: By keeping the controlled variables exactly the same, we control

them. In the example above, we would water all the experimental and control groups with precisely the same amount of water. We would also use the same soils with the same nutrient content.

• Making sure any variations are experienced equally: Some variables are difficult to control physically. In such cases, an acceptable substitute for physical control is to make sure that all plants experience the same fluctuations, thereby cancelling out the affects of the variation. In our example above, putting the plants in the same area of a room would ensure that any unavoidable temperature fluctuations in the room would be experienced by all plants in the experiment.

What is data?

Information, in the form of facts or figures obtained from experiments

or surveys, used as a basis for making calculations or drawing

conclusions Encarta dictionary

Statistics in Science

• Data can be collected about a population (surveys)

• Data can be collected about a process (experimentation)

Qualitative Data• Information that relates to characteristics or

description (observable qualities)• Information is often grouped by descriptive category• Examples

– Species of plant– Type of insect– Shades of color– Rank of flavor in taste testingRemember: qualitative data can be “scored” and evaluated

numerically

Qualitative data, manipulated numerically

• Survey results, teens and need for environmental action

Quantitative data

• Quantitative – measured using a naturally occurring numerical scale

• Examples–Chemical concentration–Temperature–Length–Weight…etc.

Quantitation

• Measurements are often displayed graphically

Quantitation = Measurement• In data collection for Biology, data must be

measured carefully, using laboratory equipment (ex. Timers, metersticks, pH meters, balances , pipettes, etc)

• The limits of the equipment used add some uncertainty to the data collected. All equipment has a certain magnitude of uncertainty. For example, is a ruler that is mass-produced a good measure of 1 cm? 1mm? 0.1mm?

• For quantitative testing, you must indicate the level of uncertainty of the tool that you are using for measurement!!

How to determine uncertainty?

• Usually the instrument manufacturer will indicate this – read what is provided by the manufacturer.

• Be sure that the number of significant digits in the data table/graph reflects the precision of the instrument used (for ex. If the manufacturer states that the accuracy of a balance is to 0.1g – and your average mass is 2.06g, be sure to round the average to 2.1g) Your data must be consistent with your measurement tool regarding significant figures.

Finding the limits

• As a “rule-of-thumb”, if not specified, use +/- 1/2 of the smallest measurement unit (ex metric ruler is lined to 1mm,so the limit of uncertainty of the ruler is +/- 0.5 mm.)

• If the room temperature is read as 25 degrees C, with a thermometer that is scored at 1 degree intervals – what is the range of possible temperatures for the room?

• (ans. +/- 0.5 degrees Celsius - if you read 15oC, it may in fact be 14.5 or 15.5 degrees)

Looking at Data

• How accurate is the data? (How close are the data to the “real” results?) This is also considered as BIAS

• How precise is the data? (All test systems have some uncertainty, due to limits of measurement) Estimation of the limits of the experimental uncertainty is essential.

Comparing Averages

• Once the 2 averages are calculated for each set of data, the average values can be plotted together on a graph, to visualize the relationship between the 2

Drawing error bars

• The simplest way to draw an error bar is to use the mean as the central point, and to use the distance of the measurement that is furthest from the average as the endpoints of the data bar

Average value

Value farthest from average

Calculated distance

What do error bars suggest?• If the bars show extensive overlap, it is likely

that there is not a significant difference between those values

Quick Review – 3 measures of “Central Tendency”

• mode: value that appears most frequently• median: When all data are listed from least to

greatest, the value at which half of the observations are greater, and half are lesser.

• The most commonly used measure of central tendency is the mean, or arithmetic average (sum of data points divided by the number of points)

How can leaf lengths be displayed graphically?

Simply measure the lengths of each and plot how many are of each length

If smoothed, the histogram data assumes this shape

This Shape?

• Is a classic bell-shaped curve, AKA Gaussian Distribution Curve, AKA a Normal Distribution curve.

• Essentially it means that in all studies with an adequate number of datapoints (>30) a significant number of results tend to be near the mean. Fewer results are found farther from the mean

Standard deviation• The standard deviation is a statistic that tells you

how tightly all the various examples are clustered around the mean in a set of data

• The STANDARD DEVIATION is a more sophisticated indicator of the precision of a set of a given number of measurements– The standard deviation is like an average deviation of

measurement values from the mean. In large studies, the standard deviation is used to draw error bars, instead of the maximum deviation.

A typical standard distribution curve

According to this curve:

• One standard deviation away from the mean in either direction on the horizontal axis (the red area on the preceding graph) accounts for somewhere around 68 percent of the data in this group.

• Two standard deviations away from the mean (the red and green areas) account for roughly 95 percent of the data.

Three Standard Deviations?

• three standard deviations (the red, green and blue areas) account for about 99 percent of the data

-3sd -2sd +/-1sd 2sd +3sd

How is Standard Deviation calculated?

With this formula!

Not the formula!• This can be calculated on a scientific calculator • OR…. In Microsoft Excel, type the following code into the cell

where you want the Standard Deviation result, using the "unbiased," or "n-1" method: =STDEV(A1:A30) (substitute the cell name of the first value in your dataset for A1, and the cell name of the last value for A30.)

You DO need to know the concept!

• standard deviation is a statistic that tells how tightly all the various datapoints are clustered around the mean in a set of data.

• When the datapoints are tightly bunched together and the bell-shaped curve is steep, the standard deviation is small.(precise results, smaller sd)

• When the datapoints are spread apart and the bell curve is relatively flat, a large standard deviation value suggests less precise results

T - TestThe Student’s t-test compares the averages and standard deviations of two samples to see if there is a significant difference between them.We start by calculating a number, t

t can be calculated using the equation:

( x1 – x2 )

(s1)2

n1

(s2)2

n2

+

t =Where:

x1 is the mean of sample 1s1 is the standard deviation of sample 1n1 is the number of individuals in sample 1x2 is the mean of sample 2s2 is the standard deviation of sample 2n2 is the number of individuals in sample 2

T-Test• used to show if the data sets occurred by chance

alone• Ex: 0.05 = 5% is chance so 95% resulted from the

experiment (good results)• Ex: 0.50 = the difference is due to chance 50% of

the time (not good)

• Degrees of Freedom used to read the t-test; total number of samples minus 2– Ex: A has 15 samples and B has 15 samples so (15+15 -

2 = 28)

Correlation Does Not Mean Causation

• Observations without experimental data can only show correlation and not causation

• Ex: smoking and lung cancer there is a high correlation between smoking and lung cancer; Does this prove smoking causes lung cancer? cannot prove that smoking is the cause unless there is experimental data to show evidence of cause

Crazy Correlations

• Shark attacks on swimmers and ice cream sales are correlated

• The number of cavities in elementary school children and vocabulary size have a strong positive correlation.

Do not worry if you do not understand how or why the test works

Follow the instructions CAREFULLY

You will NOT need to remember how to do this for your exam