Statistical AnalysisStatistical Analysis

WHY ?

Quantitative DataQuantitative Data

o Quantitative – measured using a naturally occurring numerical scale

o Examples

oChemical concentrationoTemperatureo LengthoWeight…etc.

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Qualitative DataQualitative Data

o Information that relates to characteristics or description (observable qualities)

o Information is often grouped by descriptive category

o Examples

o Species of planto Type of insecto Shades of coloro Rank of flavor in taste testingo Remember: qualitative data can be “scored” and

evaluated numericallyo

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Sampling DataSampling Data

o Don’t have enough time or resources to measure every individual in a population.

o Choose and measure a representative

sample from a population.

o Need to have a good SAMPLE SIZE in order to “believe” your data. (statistically significant)

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Can you count EVERY ONECan you count EVERY ONE

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Displaying the dataDisplaying the data

Error bars can be added to graphs to show the range of data.

This shows the highest and lowest values of the data.

MeanMean

o Another word for the average

o Calculated by summing the values and then dividing by the number of values obtained.

o Symbol: x

Statistical analysis of a sampleStatistical analysis of a sampleo Mean: is the average of data points

o Range: range is the measure of the spread of data

o Standard Deviation: is a measure of how the individual observation of data set are dispersed or spread out around the mean

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What does the standard deviation What does the standard deviation measuremeasure

o The standard deviation measures how spread out your values are.

o If the standard deviation is small, the values

are close together.

o If the standard deviation is large, the values are spread out.

o It is measured in the same units as the original data.

Standard deviationStandard deviation

Measures the spread of data around the mean.

Formula: s = √(x - x )2

BUT you do not need to remember it. You must be able to calculate it on your

calculators (or spreadsheet in the lab)

Standard DeviationStandard 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

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Standard DeviationStandard Deviation

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

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±1s (red), ±2s (green), ±3s (blue) ±1s (red), ±2s (green), ±3s (blue)

Why is it useful?Why is it useful?

o Calculate the mean of 100, 200, 300, 400, 500.

o Now let's imagine you had the values 298, 299, 300, 301, 302. Calculate the mean of these numbers.

o Although the two means are the same, the original data are very different.

The standard deviation will reflect The standard deviation will reflect this difference. this difference.

o The standard deviation of 100, 200, 300, 400, 500 is 141.4

o The standard deviation of 298, 299, 300,

301, 302 is 1.414.

o So the standard deviation of the first set of values is 100 times as big - these data are 100 times more spread out.

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Error Bars Error Bars To graphically display data, you will

use the CI to generate error bars. Error bars represent the spread

around the mean.

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Comparing Means -Comparing Means -

o What can you conclude when error bars do overlap? When error bars overlap, you can be sure the difference between the two means is not statistically significant. (Due to chance variations)

o What can you conclude when error bars do not overlap? When error bars do not overlap, you cannot be sure that the difference between two means is statistically significant. T-test is commonly used to compare these groups.

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41.6

41.6

45.9

Comparing the twoComparing the two

41.6

41.6

45.9

Why do I need both the mean and Why do I need both the mean and standard deviation?standard deviation?

o Although the standard deviation tells you about how spread out the values are, it doesn't actually tell you about the size of them.

o For example, the data 1,2,3,4,5 have the same standard deviation as the data 298,299, 300,301,302

Displaying the dataDisplaying the data

o Error bars can be added to graphs to show the standard deviation.

o This shows the spread around the mean.

Confidence Interval (CI)Confidence Interval (CI)

95% certain the mean will be found within the interval

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T-testT-test

o A common form of data analysis is to compare two sets of data to see if they are the same or different

o Null hypothesis: there is NO significant difference between.......

T-testT-test

o Calculate a value for “t”

o Compare value to a critical value (0.05 column)

o If “t” is equal to or higher than the critical value we can reject the null hypothesis.

CorrelationCorrelationo Correlation is a measure of the association between two

factors. The strength of the association between two factors can be measured.

 o An association in which all the values closely follow the

trend is described as being a strong correlation.

 o An association in which there is much variation, with

many values being far from the trend, is described as being a weak correlation.

 o A value can be given to the strength of the correlation, r.

 o r = +1 a complete positive correlationo r = 0 no correlationo r = -1 a complete negative correlation

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CorrelationCorrelation

CorrelationCorrelation

Is there a correlation between sunlight Is there a correlation between sunlight intensity and temperature?intensity and temperature?

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