SCIENCE 1101: SCIENCE, SOCIETY and the ENVIRONMENT I

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SCIENCE 1101 SCIENCE, SOCIETY and the
ENVIRONMENT I

• Lecture 4

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Statistics

• What are Statistics?

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• Statistics is a type of mathematics that allows
  us to analyze trends in data.
• Statistics is often referred to as the science of
  data..

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• Statistics (and science also) involves
• Collecting data
• Classifying data
• Summarizing data
• Organizing data
• Analyzing data
• Interpreting data

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Types of data

• Quantitative observations made on a numerical
  scale.
• Qualitative non-numerical data that can only be
  classified into one of a group of categories.

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Example

• Group Gender Age Height
  Hair Color
• M 26 510 Black
• F 30 55 Black
• 3 F 23 49 Blonde
• 4 M 26 58 Brown
• F 24 51 Red
• M 28 64 Brown
• M 21 511 Black
• F 27 510 Brown
• M 29 55 Red
• F 24 510 Blonde
• F 26 52 Brown

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Goals of statistics

• Describe data sets (populations or samples)
• To use sample data to make inferences about a
  population or group of samples.

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Two general branches of statistics

• Descriptive statistics The branch of statistics
  devoted to the organization, summarization, and
  description of data sets.
• Inferential statistics The branch of statistics
  concerned with using data to make an inference
  about a population or group of data

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Descriptive Statistics
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• Measure of Central Tendency A number that
  describes the center of a data sets distribution.
• This value is useful because it allows the
  researcher to understand where the data is most
  concentrated. This will allow researchers to
  decide what the normal condition is.

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Measure of Central Tendency

• To organize the data, place the values of a data
  set into descending order.
• 26, 30, 23, 26, 24, 28, 21, 27, 29, 24,26
• Becomes
• 30, 29, 28, 27, 26, 26, 26, 24, 24, 23, 21

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Measure of central tendency

• Mean
• Median
• Mode

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Measure of Central Tendency

• Mode
• The number that occurs most often in a data set.
• Does not need to be near the center of the data
  set.
• A data set can have more than one mode or no
  mode.

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Measure of Central Tendency

• 30, 29, 28, 27, 26, 26, 26, 24, 24, 23, 21
• What is the mode of the above data set?

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Measure of Central Tendency

• Median
• If the number of observations is odd,the value
  that occurs in the middle of a data set is the
  median.
• If the number of observations is even, the mean
  between the two middle observations is the
  median.

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Measure of Central Tendency

• 30, 29, 28, 27, 26, 26, 26, 24, 24, 23, 21
• What is the median value of the above data set?

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Measure of Central Tendency

• Mean
• Often referred to as the average
• Add up all of the observations and then divide by
  the number of observations.
• (12345)/5 3

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Measure of Central Tendency

• 30, 29, 28, 27, 26, 26, 26, 24, 24, 23, 21
• What is the mean of the above data set?

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Answers

• Mode 26
• Median 26
• Mean 25.81 26
• In this data set the Mean, Median and Mode are
  all more or less equal. This does not always
  happen though.

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Which is the best?

• This depends on the type of descriptive
  information you want.

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Measure of Dispersion

• Dispersion is a measure of the spread of a data
  set.

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Measure of Dispersion

• Range
• Gives the researcher an idea about how spread and
  diverse their data is.
• Describes the highest and lowest values in
  a data set.
• All values within a data set fall within the
  range.

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Measure of Dispersion

• 30, 29, 28, 27, 26, 26, 26, 24, 24, 23, 21
• What is the range of the above data set?
• 21 to 30

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Measure of Dispersion

• Standard Deviation and variance are also measures
  of dispersion.
• Use statistical programs to help you calculate
  these values.

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Measures of DispersionVariance and Standard
Deviation

• Standard Deviation and Variance
• Interpreted as being the average distance of the
  sample points from their center.

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Bringing it all together

• Imagine that we are studying the diameter of pine
  trees in a 200,000 acres forest with
  approximately 4,000,000 trees.
• Is it possible to measure ALL of the trees in the
  entire forest?
• AnswerNO!!

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Bringing it all together

• How can we then determine what the general tree
  diameter is of a tree in such a large forest with
  so many trees?
• We measure fewer trees (our representative
  sample), lets say 100 trees.
• Why did we pick fewer trees to measure?

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Bringing it all together

• After weve measured our trees then we try to
  determine what measure of central tendency we
  want to use.
• In this experiment well use the mean.
• How can knowing the mean tree diameter of 100
  trees tell me anything about a forest with
  4,000,000 trees?

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Bringing it all together

• Knowing the mean diameter of 100 trees can tell
  us a lot.
• However, we can never know for sure what the
  average tree diameter is in the 200,000 acre
  forest without measuring ALL of the trees. This
  would be the True Average.

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Bringing it all together

• Therefore, Since we cannot know the True Average,
  Standard Deviation helps us to know where the
  true average may lie based on our sample
  population.
• Lets say the average tree diameter for a pine
  tree in the 100 trees is 2.5 feet in diameter.

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Bringing it all together

• Based on the data in the 100 tree sample, we get
  a standard deviation of 1.3.
• Based on this we now know that the True Mean tree
  diameter may lie between 3.8 feet to 1.2 feet.

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• Take a break

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• Inferential Statistics

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Inferential Statistics

• Makes comparisons between data sets and then
  infers whether the two sets are significantly
  different from one another.
• Chance will always plays a role.
• Attempt to determine if the two means truly
  differ or is the difference just due to random
  chance.

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Inferential Statistics

• A coach wants to know if the coin flip is fixed
• Ideally, if I flip a coin there is an equal
  number of chances (probability) that either side
  will appear on top.
• To determine if this is true, I flip the coin and
  count the number of times heads comes up.

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Binomial Distribution
Animated graph
I flip the coin 10 times and these are the
probabilities that we get based on our flips. Is
ten flips enough? How about more?
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Binomial Distribution
Imagine that I flip it 50 times. Notice that the
distribution is smoother than the previous
Distribution.
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Probability

• Normal curves are useful because they allow us to
  make statistical conclusions about being a
  certain distance from the center or mean.
• 68 of all values are within one standard
  deviation, 95 within 2, and 99 are within 3
• The difficulty is knowing when to conclude an
  occurrence is not due to random chance

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Probability

• So where do we determine that the difference is
  due to random chance
• Statisticians decided that two standard
  deviations or 95 would be the cut off.
• This means that there is a 5 chance that the
  difference that you see is due to random chance.

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Probability

• Whenever a statistical test returns a probability
  value (p-value) of 0.05 (5) or less, we reject
  the null hypothesis.
• Null hypothesis states that the data fits the
  distribution.

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Probability
33 Heads
0.05 Cutoff
Back to coin toss if result of 50 flips is 33
heads And 17 tails is it part of the
distribution? NO.
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t-test

• The t-test is one of many types of inferential
  statistics that will allow you to compare two
  different groups of data and determine if they
  are statistically different.
• This test asks the question Do the data sets
  have the same distribution.

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t-test

• Imagine we want to compare the growth rate of two
  populations of fish raised on different food
  types.
• We use the t-test to compare the means and
  dispersion of the data set to determine if the
  growth rates have the same distribution.

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t-test
The t-test looks at the ratio of the differences
in the means of the two groups to the variability
of the data of the two groups.
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t-test

• This t-statistic ratio allows us to determine the
  probability for your test to determine if the
  differences between the two sets of data are due
  to random chance.
• Our t-statistic is compared to a probability
  table and our probability value is determined.
• If we get a probability of less than 0.05 we know
  that the differences between the two sets of data
  are not due to random chance and the two sets of
  data are statistically different.

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t-test

• We are always testing the Null hypothesis.
• Therefore the hypotheses we are actually testing
  under this experimental design is
• H0The growth rate of the two fish populations
  will not differ significantly.