STATISTICS A BRIEF INTRODUCTION

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STATISTICS A BRIEF INTRODUCTION
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Why Learn AboutStatistics?

• Statistics provides tools that are used in
• Quality control
• Research
• Measurements
• Sports

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In This Course

• We will use some of these tools in the lab, so
  will introduce now some
• Ideas
• Vocabulary
• A few calculations

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Variation

• There is variation in the natural world
• People vary
• Measurements vary
• Plants vary
• Weather varies

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Variation Cont

• Variation among organisms is the basis of natural
  selection and evolution

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Example

• 100 people take a drug and 75 of them get better
• 100 people dont take the drug but 68 get better
  without it
• Did the drug help?

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Variability Is A Problem

• There is variation in peoples response to the
  illness
• There is variation in peoples response to the
  drug
• So its difficult to figure out if the drug
  helped
• Statistics helps with this type of problem

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Statistics

• Provides mathematical tools to help arrive at
  meaningful conclusions in the presence of
  variability

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Statistics Cont

• Might help researchers decide if a drug is
  helpful or not
• This is a more advanced application of statistics
  than we will get into

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Statistics Cont

• Lets begin with some basic vocabulary

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Definitions

• Population
• Entire group of events, objects, results, or
  individuals, all of whom share some unifying
  characteristic

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Populations

• Examples
• All of a persons red blood cells
• All the enzyme molecules in a test tube
• All the college graduates in the U.S.

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Sample

• Sample Portion of the whole population that
  represents the whole population

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Sample Cont

• Example It is virtually impossible to measure
  the level of hemoglobin in every cell of a
  patient
• Rather, take a sample of the patients blood and
  measure the hemoglobin level

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Population
SAMPLE
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More About Samples

• Representative sample sample that truly
  represents the variability in the population --
  good sample

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Representative Sample

• A sample is random if all members of the
  population have an equal chance of being drawn
• A sample is independent if the choice of one
  member does not influence the choice of another
• Samples need to be taken randomly and
  independently in order to be representative

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Sampling

• How we take a sample is critical and often
  complex
• If sample is not taken correctly, it will not be
  representative

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Example

• How would you sample a field of corn?
• Think about how to get a good sample

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Variables

• Variables
• Characteristics of a population (or a sample)
  that can be observed or measured
• Called variables because they can vary among
  individuals

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Variables

• Examples
• Blood hemoglobin levels
• Activity of enzymes
• Test scores of students

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Variables Cont

• A population or sample can have many variables
  that can be studied
• Example
• Same population of six year old children can be
  studied for
• Height
• Shoe size
• Reading level
• Etc.

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Data

• Data Observations of a variable (singular is
  datum)
• May or may not be numerical
• Examples
• Heights of all the children in a sample
  (numerical)
• Lengths of insects (numerical)
• Pictures of mouse kidney cells (not numerical)

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Always Uncertainty

• Even if you take a sample correctly, there is
  uncertainty when you use a sample to represent
  the whole population
• Various samples from the same population are
  unlikely to be identical
• So, need to be careful about drawing conclusions
  about a population, based on a sample there is
  always some uncertainty

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Sample Size

• If a sample is drawn correctly, then, the larger
  the sample, the more likely it is to accurately
  reflect the entire population
• If it is not done correctly, then a bigger sample
  may not be any better
• How does this apply to the corn field?

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Inferential Statistics

• A branch of statistics
• Wont talk about it much
• Deals with tools to handle the uncertainty of
  using a sample to represent a population
• Helps with problems like the drug study,
  mentioned earlier

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Descriptive Statistics

• Chapter 11 in your textbook
• Descriptive statistics is one area within
  statistics
• The is the type of statistics we will use

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Descriptive Statistics

• Provides tools to DESCRIBE, organize and
  interpret variability in our observations of the
  natural world

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Example Problem

• In a quality control setting, 15 vials of product
  from a batch are tested. What is the sample?
  What is the population?
• In an experiment, the effect of a carcinogenic
  compound was tested on 2000 lab rats. What is
  the sample? What is the population?

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Example Problem Cont

• A clinical study of a new drug was tested on
  fifty patients. What is the sample? What is the
  population?

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Answers

• 15 vials, the sample, were tested for QC. The
  population is all the vials in the batch.
• The sample is the rats that were tested. The
  population is probably all lab rats.
• The sample is the 50 patients tested in the
  trial. The population is all patients with the
  same condition.

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Example Problem

• An advertisement says that 2 out of 3 doctors
  recommend Brand X.
• What is the sample? What is the population?
• Is the sample representative?
• Does this statement ensure that Brand X is better
  than competitors?

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Answer

• Many abuses of statistics relate to poor
  sampling. The population of interest is all
  doctors. No way to know what the sample is. The
  sample could have included only relatives of
  employees at Brand X headquarters, or only
  doctors in a certain area. Therefore the
  statement does not ensure that the majority of
  doctors recommend Brand X. It certainly does not
  ensure that Brand X is best.

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Describing Data Sets

• Draw a sample from a population
• Measure values for a particular variable
• Result is a data set

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Data Sets

• Individuals vary, therefore the data set has
  variation
• Data without organization is like letters that
  arent arranged into words

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Data Sets Cont

• Numerical data can be arranged in ways that are
  meaningful or that are confusing or deceptive

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Descriptive Statistics

• Provides tools to organize, summarize, and
  describe data in meaningful ways
• Example
• Exam scores for a class is the data set
• What is the variable of interest?
• We can summarize the data with the class
  average, what does this tell you?

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Descriptive Statistics Cont

• A measure that describes a data set, such as the
  average, is sometimes called a statistic
• Average gives information about the center of the
  data

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Median And Mode

• Two other statistics that give information about
  the center of a set of data
• Median is the middle value
• Mode is most frequent value

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Measures OfCentral Tendency

• Measures that describe the center of a data set
  are called Measures of Central Tendency
• Mean, median, and the mode

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Hypothetical Data Set

• 2 5 6 7 8 3 9 3 10 4 7 4 6 11 9
• Simplest way to organize them is to put in
  order
• 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11
• By inspection they center around 6 or 7

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Mean

• Mean is basically the same as the average
• Add all the numbers together and divide
• by number of values
• 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11
• What is the mean for this data set?

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Nomenclature

• Mean 6.3 read X bar
• The observations are called X1, X2, etc.
• There are 15 observations in this example, so the
  last one is X15
• Mean Xi
• ___n
• Where n number of values

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Example

• Data set
• 2 3 3 4 5 6 7 8 9
• What is the mode?
• What is the median?

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Mean Of A Population Versus The Mean Of A Sample

• Statisticians distinguish between the mean of a
  sample and the mean of a population
• The sample mean is
• The population mean is
• It is rare to know the population mean, so the
  sample mean is used to represent it

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Dispersion

• Data sets A and B both have the same average
• A 4 5 5 5 6 6
• B 1 2 4 7 8 9
• But are not the same
• A is more clumped around the center of the
  central value
• B is more dispersed, or spread out

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Measures Of Dispersion

• Measures of central tendency do not describe how
  dispersed a data set is
• Measures of dispersion do they describe how much
  the values in a data set vary from one another

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Measures Of Dispersion

• Common measures of dispersion are
• Range
• Variance
• Standard deviation
• Coefficient of variation

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Calculations Of Dispersion

• Measures of dispersion, like measures of central
  tendency, are calculated
• Range is the difference between the lowest and
  highest values in a data set

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Calculations Of Dispersion Cont

• Example
• 2 3 3 4 4 5 6 6 7 7 8 9 9 10 11
• Range 11-2 9 or, 2 to 11
• Range is not particularly informative because it
  is based only on two values from the data set

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Calculating Variance And Standard Deviation

• Variance and standard deviation measure of the
  average amount by which each observation varies
  from the mea
• Example
• 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm
• This is a data set, the lengths of 8 insects

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Calculating Variance And Standard Deviation

• 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm
• The mean is 7 cm
• How much do they vary from one another?
• Intuitively might see how much each point varies
  from the mean
• This is called the deviation

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Calculation OfDeviations From Mean

• 4cm 5cm 6cm 7cm 7cm 7cm 9cm 11cm
• Value-Mean Deviation
• in cm
• (4-7) - 3
• (5-7) - 2
• (6-7) - 1
• (7-7) 0
• (7-7) 0
• (7-7) 0
• (9-7) 2
• (11-7) 4

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Calculation OfDeviations From Mean Cont

• Value-Mean
  Deviation
• (in cm)
• (4-7) - 3
• (5-7) - 2
• (6-7) - 1
• (7-7) 0
• (7-7) 0
• (7-7) 0
• (9-7) 2
• (11-7) 4
• Sum of deviations 0

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Calculation OfDeviations From Mean Cont

• Sum of the deviations from the mean is always
  zero
• Therefore, cannot use the average deviation to
  describe the dispersion in the data set
• Therefore, mathematicians decided to square each
  deviation so they will get positive numbers

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Calculation OfDeviations From Mean Cont

• Value-Mean Deviation Squared Deviation
• (in cm)
• (4-7) - 3 9 cm2
• (5-7) - 2 4 cm2
• (6-7) - 1 1 cm2
• (7-7) 0 0
• (7-7) 0 0
• (7-7) 0 0
• (9-7) 2 4 cm2
• (11-7) 4 16 cm2
• total squared deviation sum of
  squares 34 cm2

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Variance

• Total squared deviation (sum of squares) divided
  by the number of measurements
• 34 cm2 4.25 cm2
• 8

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Standard Deviation

• Square root of the variance
• 4.25 cm2 2.06 cm
• Note that the SD has the same units as the data
• Note also that the larger the variance and SD,
  the more dispersed are the data

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Variance And SDOf Population Vs Sample

• Statisticians distinguish between the mean and SD
  of a population and a sample
• The variance of a population is called sigma
  squared, s2
• Variance of a sample is S2

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Variance And SDOf Population Vs Sample Cont..

• The standard deviation of a population is called
  sigma, s
• Standard deviation of a sample is S or SD

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Standard Deviation Of A Sample
(Xi - )2 n -1
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Example Problem

• A biotechnology company sells cultures of E.
  coli. The bacteria are grown in batches that are
  freeze dried and packaged into vials. Each vial
  is expected to have 200 mg of bacteria. A QC
  technician tests a sample of vials from each
  batch and reports the mean weight and SD.

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Example Problem Cont

• Batch Q-21 has a mean weight of 200 mg and a SD
  of 12 mg. Batch P-34 has a mean weight of 200 mg
  and as SD of 4 mg. Which lot appears to have
  been packaged in a more controlled fashion?

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Answer

• The SD can be interpreted as an indication of
  consistency. The SD of the weights of Batch P-34
  is lower than of Batch Q-21. Therefore, the
  weights for vials for Batch P-34 are less
  dispersed than those for Batch Q-21 and Batch
  P-34 appears to have been better controlled.

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FrequencyDistributions

• So far, talked about calculations to describe
  data sets
• Now talk about graphical methods

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The Weights Of 175 Field Mice

• (in grams)
• 19 22 20 24 22 19 27 20 21 22 20 22 24 24 21 2
  5 19 21 20 23 25 22 19 17 20 20 21 25 21 22 27 22
  19 22 23 22 25 22 24 23 20 21 22 23 21 24 19 21 22
  22 25 22 23 20 23 22 22 26 21 24 23 21 25 20 23 2
  0 21 24 23 18 20 23 21 22 22 25 21 23 22 24 20 21
  23 21 19 21 24 20 22 23 20 22 19 22 24 20 25 21 22
  22 24 21 22 23 25 21 19 19 21 23 22 22 24 21 23 2
  2 23 28 20 23 26 21 22 24 20 21 23 20 22 23 21 19
  20 26 22 20 21 22 23 24 20 21 23 22 24 21 23 22 2
  4 21 22 24 20 22 21 23 26 21 22 23 24 21 23 20 20
  21 25 22 20 22 21 21 23 22

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The Weights Of 175 Field Mice Cont

• This table of raw data is hard to interpret
• Begin by making a frequency table

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FrequencyDistribution Table Of The Weights Of
Field Mice

• Weight Frequency
• (in grams)
• 1
• 11
• 25
• 25
• 34
• 40
• 27
• 19
• 10
• 4
• 2
• 1

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Frequency Table

• Tells us that most mice have weights in the
  middle of the range, a few are lighter or heavier
• The word distribution refers to a pattern of
  variation for a given variable

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Frequency Table Cont

• It is important to be aware of patterns, or
  distributions, that emerge when data are
  organized by frequency
• The frequency distribution can be illustrated as
  a frequency histogram

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Frequency Histogram

• X axis is units of measurement, in this example,
  weight in grams
• Y axis is the frequency of a particular value
• For example, 11 mice weighed 19 g
• The values for these 11 mice are illustrated as a
  bar

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Frequency Histogram Cont

• Note that when the mouse data were collected, a
  mouse recorded as 19 grams actually weighed
  between 18.5 g and 19.4 g.
• Therefore the bar spans an interval of 1 gram

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FIRST FOUR BARS
F R E Q U E N C Y
17 18 19 20
WEIGHTS IN GRAMS
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Constructing AFrequency Histogram

• Divide the range of the data into intervals
• It is simplest to make each interval (class) the
  same width
• No set rule as to how many intervals to have
• For example, length data might be 1-9 cm, 10-19
  cm, 20-29 cm and so on

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Constructing AFrequency Histogram Cont

• Count the number of observations that are in each
  interval
• Make a frequency table with each interval and the
  frequency of values in that interval
• Label the axes of a graph with the intervals on
  the X axis and the frequency on the Y axis

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Constructing AFrequency Histogram Cont

• Draw in bars where the height of a bar
  corresponds to the frequency of the value
• Center the bars above the midpoint of the class
  interval
• For example, if the interval is 0-9 cm, then the
  bar should be centered at 4.5 cm

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NormalFrequency Distribution

• If weights of very many lab mice were measured,
  would likely have a frequency distribution that
  looks like a bell shape, also called the normal
  distribution

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NormalDistribution
F R E Q U E N C Y
WEIGHT
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Normal Distribution

• Very important
• Examples
• Heights of humans
• Measure same thing over and over, measurements
  will have this distribution

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Calculations AndGraphical Methods

• Related
• The center of the peak of a normal curve is the
  mean, the median and the mode
• Values are evenly spread out on either side of
  that high point

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Calculations AndGraphical Methods Cont

• The width of the normal curve is related to the
  SD
• The more dispersed the data, the higher the SD
  and the wider the normal curve
• Exact relationship is in text, not go into it
  this semester

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Example Problem

• A technician customarily performs a certain
  assay. The results of 8 typical assays are
• 32.0 mg 28.9 mg 23.4 mg 30.7 mg
• 23.6 mg 21.5 mg 29.8 mg 27.4 mg
• If the technician obtains a value of 18.1 mg,
  should he be concerned? Base your answer on
  estimation.
• Perform statistical calculations to see if the
  answer if out of the range of two SDs.

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Answer

• The average appears to be in the mid-twenties and
  hovers around 5. Therefore, 18.1 mg appears a
  bit low.
• Mean 27.16 mg, SD 3.87 mg. The mean 2SD
  is 19.4 mg, so 18.1 mg appears to be outside the
  range and should be investigated