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Univariate Statistics

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LIR 832

Class #2

September 15, 2008

- Fundamental Problem in Statistics: Learning about populations from samples
- Describing Data Compactly:
- How we might describe data, why compactness matters.
- Measures of Central Tendency (what are they, when to use them)
- Measures of dispersion

- Probability Distributions:
- As samples are hopefully random draws from populations, we need to understand the likelihood of drawing samples. This leads us to a review of some basic probability distributions.

- Inference from Samples to Populations:
- Sampling Distributions and the Central Limit Theorem
- Estimation
- Hypothesis Testing

- Populations and Samples: Generally wish to know about populations
- What is a population?
- How do we count a population?
- What types of populations would you be concerned with in your professional life?

- Use of Samples to Learn about Populations
- What is a sample?
- representative sample
- random sample
- convenience sample

- Why use samples rather than populations?
- less time consuming to collect
- less expensive to collect
- often more accurate than census
- population may not exist at the time data is collected

- Samples are affected by randomness, two samples drawn from a population are unlikely to be identical (sampling variability) and neither is an exact reproduction of the population.
- What is meant by random?
- An event is random if, despite knowing all of the possible outcomes in advance, we are not able to exactly predict a particular outcome.
- experiment: M&M issue
- Samples are, to some degree, random
- different samples produce different estimates
- sample mean may be different than (but close to) population mean

- What is meant by random?

- Since sample is not an exact reproduction of the population, we need to allow for sampling variability in using samples to tell us about populations.
- What types of HR/IR issues might involve the use of samples?

- We are interested in a training program which is supposed to improve productivity. It would be very expensive to implement throughout a firm, particularly if it doesn’t work. Instead, we set up an experiment in which we try the program on a sample of employees at a single location (this may be called a pilot program).

- We experiment with a pilot program and find that productivity rose by 2% Our problem in using the pilot (sample): if we replicate the pilot throughout the firm is it reasonable to believe that:
- we will get a 2% boost in productivity, or
- This could this just be the result of getting a “good” sample (Folks who happened to respond favorably to the program).

- Our core problem in using samples is distinguishing between systematic effects of programs and chance outcomes

- You run human resources for a large low wage manufacturing plant. Your firm has established that there should be a 2.5% turnover rate per month and has a policy that turnover rates above 2.5% are evidence of ineffective human resource programs.
- You have kept your turnover rate at 2.4% per month for the last year and one half. Last month and this month that rate has shot up to 3.2%. Is this evidence that you are not doing your job as a human resource manager?

- Answer: The numeric representation of the characteristics of an individual, object or experiment.
- You, as an individual, provide multi-dimensional data:
- Quantitative:
- Age
- Height
- Your pre-class views on LIR 832

- Qualitative:
- Gender
- Educational Attainment
- Occupation

- Economic Data:
- Income
- Debt
- Expected Income on graduating from this program.

- Quantitative:

- You are multi-dimensional. We may be very interested in the relationship between your characteristics:
- Gender and Educational Attainment with expected income
- Blood pressure with age

- Data can also be collected on plants, establishments and firms as well as by political or geographic entity.
- Firm: Revenue, Operating Costs, Debt to Equity Ratio, Number of Employees, Number of Locations, Distribution of Occupations, Presence of Program to Encourage Diversity in the Labor Force

- We are typically interested in variables, data which vary across ‘individuals’ or ‘units of observation.
- age, gender, income vary across individuals in this class
- revenue, profit (and so much more) varies across divisions of General Motors
- Some characteristics do not vary in a given data set. For example, all of you will have the same level of educational attainment, a masters degree, when you graduate. Such characteristics are called constants. In contrast, level of education will vary in a survey of MSU graduates

- Some of the difference in individual outcomes in variables is systematic (predictable), some is random.
- There is systematic difference in earnings by gender & education
- Still, if we take women graduates of this program with three years of post graduation experience there will be considerable differences in annual earnings. Much of this is random in the sense that we could not predict it in advance.

- Qualitative or nominal data:
- The numeric values indicate qualitative states but does not indicate rank or any arithmetic relationship.
- northeast = 1/midwest = 2/south = 3/west = 4
- White =1/Black =2/Asian-Pacific Islander = 3

- The numeric values designate a qualitative outcome but they could be changed around without any loss of information. E.G.:
- northeast = 20/midwest = 100/south = -3/west = 41,298
- White =1/Black =20/Asian-Pacific Islander = 3000

- The numeric values indicate qualitative states but does not indicate rank or any arithmetic relationship.

- Ordinal or ranked data:
- Likert scale: (subjective)
- 2 = strongly dislike/ -1 = dislike/ 0 = like/ 1 = strongly like

- Income: (objective)
- 0 - income under $4,999 per year
- 1 - income from $5,000 to $9,999
- 2 - income from $10,000 to $14,999

- The values of the data designate a ranking or order and we can tell if the outcomes are =, > or < but we cannot perform arithmetic operations.

- Likert scale: (subjective)

- Cardinal Data (also called interval or ratio data): we can apply arithmetic operations to this data (=, >, < +, -, x, ÷). There can be values of 0 as well as negative values.
- weekly earnings
- age
- number of absences last week

- The Problem of Compactly Describing Populations
- Ages of students in the class (33 observations on age from 2004):
2537262551 37272139222327222223252522222423322325272127232223243122

- Ages of students in the class (33 observations on age from 2004):

- Possible first step to summarize: Calculate a range.
- Example: Class Age Data
- Range: 30 (from 21 to 51)

- Example: Frito-Lay HR Metrics
- Range: 27.64% (from 1.00% to 28.64%)

- Example: Class Age Data

- Another possible solution: Histogram
- One option: Create table of absolute and relative frequencies.
Age Distribution of LIR 832 on Wednesday August 29, 2005

AGEAbsolute FrequencyRelative Frequency

20 to 2417.515 (= 17/33)

25 to 2910.303 (= 10/33)

30 to 342.061 (= 2/33)

35 to 393.091 (= 3/33)

40 to 4400

45 to 4900

50 to 541.030 (= 1/33)

- One option: Create table of absolute and relative frequencies.

- Another option: Graph histogram

- Graph of Frito-Lay HR Metrics:

- Measures of central tendency: (Mode, Median & Mean)
- Extremely compact, a single measure is used to represent useful information about possibly large bodies of data.
- Examples:
- Mean unemployment rate
- Average turnover (number or rate)
- Average earnings
- Health care program most commonly chosen by employees.

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- Mode - the most common or frequent outcome
- In the class age data the mode is:_________
- Not necessarily unique
- Can be computed for all types of data
- Important method in advanced statistical analysis (maximum likelihood estimation)

- Median -the middle (or central) observation if observations are ranked from largest to smallest
- In the class age data the median is:______
- Median is unique
- Can be computed for cardinal and ordinal data
- Uses only the central observation
- Insensitive to outliers.
- Median (w/ 51 year old) = median (w/out) = 24

- Mean - average observation accounting for value of observations. It is the sum of all observations divided by the number of observations.
- Mean - sum of all observations divided by the number of observations

- Class Age: Population Mean for 33 Students: μage = 868/33 = 26.3
- All cardinal data has a mean (but ordinal and qualitative doesn’t)
- Uses all of the data in the calculation.
- Only one mean
- Represents the balance point of the data
- Sensitive to outliers.
- Remove our 51 year old from the data: μ = 25.8

- Q: When Should We Use a Mode, a Median or a Mean?
- A: Our goal is to give the reader an idea of the “typical” outcome
- Mode: lovely when there are a few possible outcomes and you want to indicate which is most popular.
- Which medical plan among five is chosen most frequently by our employees?
- Not so good when you have 10,000 outcomes and there are few duplicates. The mode doesn’t really tell you very much (see our class age and Frito-Lay HRM data).
- Model approaches lead fairly quickly to pie chart presentations as these quickly summarize information on matters such as “portion participating”.

- Mode: lovely when there are a few possible outcomes and you want to indicate which is most popular.

- Mean: good because is summarizes so much data but can be sensitive to outliers and can average across important distinctions.
- Consider two samples of annual income: one with Bill Gates, one without Bill Gates.
- Consider the following data on five employees blood pressure.
- NormalHigh
- 80 150
- 83 170
- 75

- What is the average blood pressure among our employees? If 120 is a cut off between normal and high blood pressure, does the mean help us to understand health issues among employees in our firm?

- Median: good because it is invariant to outliers, but it doesn’t use data very efficiently. Unless the data is skewed, the mean uses a lot more data efficiently than medians and we can develop measures of dispersion for means.

- Geometric or Harmonic Means:
- A type of mean that we won’t use very much in this course
- Suppose we have a problem in which we are calculating the return to an investment over time in which the return is
- year 15%
- year 210%
- year 312%

- So your equation for calculating your return on principle would be: principle*(1.05)*(1.10)*(1.12)

- What is the mean rate of return? If we used our arithmetic mean, we would get: (5% + 10% + 12%)/3 = 9%.
- This is wrong because it doesn’t allow for compounding. The correct method of calculating the mean is

- This is pretty close to the arithmetic mean, but if we averaged: 10%, 11%, 12%, 13%, 14%, 15%, 100% interest rates:
- arithmetic mean: 23.7%
- Geometric mean: 26.2%=

- The geometric mean is the root of the number of values you multiply to get your final result

- A story of labor relations specialists and TV reporters:
- While the average may be the same, obviously the two jobs are not the same in terms of pay. So we need to have a statistic that can tell us whether the numbers are close together, as with HR managers, or spread out, as with TV Broadcasters.

- Another example. Average Temperatures in East Lansing and Mercury are similar 65○F vs 63.7○F. However, the dark side of mercury is close to absolute zero Kelvin and the sun side is around 350○F. The mean temperature in E.L. conveys much more useful information than does the mean for mercury.

- Dispersion: What is it, how do we measure it?
- Dispersion is very important if we are using samples to learn about populations. Randomness in sampling results in sample means being dispersed around the population mean. As a result, we don’t believe that sample statistics exactly reproduce population characteristics. If we are going to use samples, we need to figure out how to handle this dispersion (sampling variability).

- Most people are comfortable with measures of central tendency, but not with dispersion. Witness Human Resource Dashboards.

- Issue:
- How close is a typical observation to the mean?
- How much information does our mean contain?

- Variance and Standard Deviation:
- Dispersion is more about distance than direction. Move to using a distance measure
- Need a measure which is
- A measure of distance (don’t want sign)
- In the same units as the underlying data
- Mathematically tractable

- All Very Nice, so what do we do with Standard Deviation?
- We could work with the empirical rule or Chebychev’s (or is it Tschebychev’s) inequality
- This would give us a taste of what we will learn when we work with the normal distribution and the Central Limit Theorem. Since we are moving fast, we will put these aside and wait until we get there next week.
- For the moment, we have a couple measures of dispersion, we don’t really know what they mean.

- Return to our fundamental problem: using samples to learn about unobserved populations
- Samples are random (hopefully) draws from populations, we need to understand the likelihood of drawing a particular samples.
- Our understanding of randomness is based on probability theory. Probability theory is used to understand events where all possible outcomes are known in advance, but where it is not possible to predict the actual outcome with certainty.
- Drawing a particular hand in a poker game
- How many employees will be absent tomorrow
- The profitability of your department in the next quarter.

- Our understanding of randomness is based on probability theory. Probability theory is used to understand events where all possible outcomes are known in advance, but where it is not possible to predict the actual outcome with certainty.

- Experiment: An experiment: an action whose outcome cannot be known with certainty in advance
- Flip a coin three times and count the number of heads
- The number of employees absent on a given day.
- The consequences of ...

- The number of heads on three flips
- The number of employees absent

- A listing of events and the likelihood that those events will occur.
- Flip a coin three times and count the number of heads:
- HeadsProbability:
- 01/8
- 13/8
- 23/8
- 31/8

- Note: 0 ≤P ≤1

- Problem: Our plant has 1,500 employees. We want to learn about their views on a variety of issues. The survey a consultant has designed takes 30 minutes and that is too long a survey to give to the workforce of the entire plant. Instead, we are going to sample 5%, 1/20th, of the plant workforce.
- We would like the sample to be representative and for every employee to have an equal chance of being chosen to participate in the sample. How can we do this?

- The probability distribution in which all individuals are equally likely to be chosen is a uniform distribution
- Every outcome is equally likely
- Rolling a fair die
- Commonly used in lotteries and to draw simple random samples

- With integer (discrete) data, the formula for the probability of any one person being drawn is:
- Why add 1?
- Consider a problem in which we have a distribution of the hourly earnings of employees (who earn between 5 and10 per hour). If we just calculate the number of points as : 10- 5 we get 5 or 1/5 probability; But there are 6 points: 5 6 7 8 9 10 as we include the bottom point. So we need to add 1 back in.

- Return to our problem using statistical software: (USE MINITAB IN CLASSROOM).
- We have a list of all students enrolled in the SLIR currently.
- Use MINITAB to randomly assign a number between 0 and 100 (or 0 and 1) at random. Each number has an equal likelihood of being assigned.
- Now chose an interval that contains 5% of all the numbers between 0 and 100. For example, 0 - 5 or 60 - 65. This should actually be done in advance.
- The employees in this group are your chosen 5%.

- Congratulations, you have now chosen a random 5% sample!

- Staffing a Call Center: How Many Employees When the Number of Calls Coming in a Random?
- Example: We oversee a benefits call center. Call center personnel indicate that a typical call takes 50 minutes to field. Personnel also get a 10 minute break every hour to be able to collect their thoughts, get to a bathroom, and so on. So our typical employee can handle one call per hour. Our records indicate that the call center averages 20 calls per hour.
- We are concerned about the quality of the service provided by the call center. Quality has many dimensions, but a critical dimension is whether employees are able to get through when they call. How many employees do we need to be certain we will not turn away more than 1 in 10 calls? More than 1 in 20 calls?

- Poisson - developed to predict the number of events occurring at a particular time or in a particular space:
- Cars arriving at a toll booth
- Dents in a one meter square of sheet metal
- Deaths due to kicks in the head by a horse in a Prussian cavalry division
- Number of calls arriving at a call center in an hour (related to manning)

Some Mathematical Argot

e, the natural base, is 2.71828182845905. So, e5 = 2.718281828459055 = 148.41

x! = x*(x-1)*(x-2)*(x-3).....3*2*1

5! = 5*4*3*2*1

- Learning to Use the Poisson Formula with a Simple Problem:
- Average 3 calls per hour, calculate the probability of up to 10 calls

- You calculate p(3), p(4) & p(5) (Divide the classroom)

- Now calculate the probability of no more than 1, 2, 3 ..... phone calls (This is referred to as a cumulative probability)
- P(1) = .1493
- P(1 or 2 ) = .1493 + .2240
- P(1 or 2 or 3) = ?

- A Problem of Discrimination, or Is It?
- XYZ firm is going through a layoff at a plant. Thirty-eight percent of employees are age 40 or older. The firm downsizes and lays off 100 employees. 47 of those employees are age 40 or older. Does this firm have a problem with the ADEA?

- Legal standard: A layoff provides prima facia evidence of discrimination if the observed outcome, an excessive proportion of older workers being laid off, has less than a 5% probability of happening by chance.
- An excess number of older workers have being laid off. What is the likelihood that 47 percent of the laid off workers would be 40 or older absent some intent to reduce the proportion of older workers in the labor force?

- Binomial:We can model our layoff as a binomial event, you are either laid off or not laid off.
- underlying a binomial is an event with only two outcomes.
- typically interested the likelihood of a set of multiple outcomes.
- Proto-typical example: The likelihood of getting X heads on flipping a coin Y times. Consider the likelihood of flipping a coin 3 times. What is the probability distribution for a coin which has a 75% chance of coming up heads?

- Example: You are at a firm in which 38% of employees are age 40 or older. The firm downsizes and lays off 100 employees. 47 of those employees are age 40 or older. Does this firm have a problem with the ADEA?
- Probabilistic issue: each layoff decision is a yes/no decision similar to a coin toss. Formally, we are asking the equivalent of the following problem. Consider a coin which turns up heads 38% of the time (not a fair coin). What is the likelihood on 100 flips that we will get 47 heads?

- Survival or Hazard Function – Exponential Distribution: (Time to failure)
- originally developed to predict the expected length of life of a lightbulb
- How long an employee will be absent from work with an injury.
- How long a machine will run prior to needing to be adjusted.
- How long an employee will remain with a firm or in a particular job.
- Because this involves time, which is a continuous variable, we turn to this a little later in our discussion of univariate statistics.

- To this point we have worked with random variables in which outcomes were discrete (integer) and which each integer outcome was associated with a probability.
- Our binomial problem: What is the likelihood that out of a group of fifty employees, eight will resign in the next month?

- Many of the numbers we are concerned with are continuous
- turnover rates
- time to resignation or time itself
- earnings (too many to treat as discrete)

- With continuous numbers, we can find a number between any two other numbers. For example, 1.5 falls between 1 and 2 while 1.27 falls between 1 and 1.5 (as do many other numbers).

- Example: Consider a problem in which we have a distribution of the hourly earnings of employees (who earn between 5 and10 per hour). To this point, we have been working with integers, its as if we assumed that people earned $5, $6, $7, $8, $9, or $10. If wages were uniformly distributed, then 1/6 employees would be at each dollar amount. So if we asked what proportion of employees earned less than $6, it would be 1/6.

- Now consider the more realistic situation, in which employees can earn any amount between $5 and $6 – lets still assume that earnings are uniformly distributed
- How might we calculate the probability that the wage falls between $5.00 and $6.00?
- P(5.00 x 6.00) = 1/5th of the area between 5 and 10 = .20 or 20%

- We are now working with areas rather than with points.
- Rather than use probabilities P(X) where we talk about the P(5.5) or P(6), we use:
- Points no longer have any probability value (weight)
- P(5.50) = 0 = p(6.0) = P(X= x for any x)

- This occurs because, there are an infinity (∞)of numbers between 5 and 7. If we divide 1, the value of 5.5 by infiinity, the outcome is zero.
- 1/∞ = 0

- So, with continuous numbers, no single point has any value. Instead, we need to think about area.

- Why are we learning about the normal?
- The “bell curve” was first noticed when 18th century astronomers noticed that errors in their predictions of the positions of plants and other heavenly bodies tended to cluster symmetrically around the mean. A graph of the errors had a bell shape.
- In the 19th century a Belgian astronomer observed that this “law of error” also applied to many human phenomena such as the chest sizes of more than 5,000 Scottish soldiers (the mean was 40")

- As a matter of statistics, the bell curve is assured to arise whenever some variable, such as human height, is determined by lots of little causes such as genes, health and diet (New Yorker, page 86, January 24, 2005).

- Good description of natural and social phenomena.
- Natural phenomena
- neck and arm lengths
- Heights

- Social phenomena
- grades in a large class
- weekly earnings (log of)

- Samples taken from large populations
- Central Limit Theorem tells us that the means of samples of 30 or more observations are normally distributed around the population mean.

- Natural phenomena

- Characteristics of the Normal
- Symmetric around its mean
- mean = media = mode
- concentrated close to the mean:
- 68% of observations are ± σ (within one standard deviation) of the mean
- 95.44% of observations are within 2 standard deviations of the mean.

- Suddenly, we are using Standard Deviations for a purpose. More to follow.

- Symmetric around its mean

- If x (or y or w or any random variable) is normal, then we write it out in shorthand as
- x ~ N(μ, σ2)
- Where μ is the population mean and σ2 is the population variance

- So if we have a normal with mean 3 and variance 4, we can write that normal out as (using w as the designation of our random variable)
- W ~ N(3, 4)

- The Z distribution:
- Fortunately we have tables for a particular normal distribution
- z ~ N(0,1) normally distributed mean 0 variance 1

- Pass Out Z-Table Here.

- Using the Table for Z values:
- P(z ≥1.5) where z~N(0,1)?
- Translate:
- what is the probability of observing a z at least 1.5 standard deviations above the mean (>= 1.5) if the distribution of z is normal 0,1?
- If we chose 100 random values from a standard normal distribution, what proportion would be greater than or equal to 1.5?
- Use of the Table: Find 1.5 on the vertical scale, find 0.00 on the horizontal scale outcome.

- P(z ≥1.50) = .0668 or 6.68% so the body of the table provides the probability of an event

- A problem: The scores for an employment exam are normally distributed with a mean of 500 and a variance of 5625. Your boss only wants to hire individuals scoring more than 650 on this exam. What is the likelihood of observing a score of more than 650?
- Note: the random variable GRE ~ N(500,5625) and has standard deviation of 75

- What is the likelihood of observing a score of more than 650?
- Pr(GRE ≥ 650)
- Pr((GRE - μ)/σ ≥(650 - 500)/75))
- Pr(z ≥(150/75))
- Pr(z ≥2) = .0228 or 2.28%

- If our boss wants us to hire 10 employees with scores of 650 or more, how many exams must we give to be reasonably sure of getting ten qualified prospects?

- Our statistical issue: If we wish to use samples to say useful things about unobserved populations, then we need to know the relationship between populations and the samples they produce
or

- What is the relationship between the characteristics of a population and the mean of a sample drawn from that population?
- Formally: if x comes from a population with mean μ and variance σ2, how is distributed?

- Clarifying our thinking about samples:
- How many samples are there in a population?
- Population of 10 and we take a sample of 3. There are
- 10C3 = 10!/3!*7! (Aka the combination of ten things taken three at a time)
- Where 3! = 3*2*1
- 7! = 7*6*5*4*3*2*1

- Samples = 120 samples

- How many samples are there in a population?

- Example of the number of samples which can be obtained from a population:
- Population of 5: sample size 3:
- We have five individuals age 22, 24, 26, 28, and 30. Take samples of 3.
- What is the mean and variance of the population?
- How many samples of three are there?
- What is the distribution of mean ages among these samples of three?

- What is the mean and variance of the population?
- μ = 26
- σ2 = 8

- How many samples of three are there.?
- 5C3= 5!/3!*2! = 10

- What is the distribution of mean ages among these samples of three?

- There are many samples which may be drawn from a given population. We typically draw a single sample, but if we drew systematically, we could draw a very large number of samples. Hence we can talk about the distribution of x-bar.
- Sample Means are close to μ, but they are not necessarily equal to it.

- Different samples produce different means. Samples do not exactly reproduce population characteristics. This is known as sampling variability. Some texts refer to this as sampling error but this is a misleading term.
- The variance of the sample means is much smaller than the variance of the population data.
- The single sample is a source of profound confusion. Because we draw but one, it doesn’t look random, we don’t see the sampling variability and don’t always understand why the sample is not the ‘correct’ value of the population.

- This result is without reference to the distribution of the underlying population.
- Why is this important? Because if we don’t know the mean and variance of the population, we probably don’t know the distribution. But with the CLT, we can still do a lot in discussing the relationship of the sample to the population.

- As n becomes large, variance around μ becomes small. This is sensible since observations which are far above the mean tend to get averaged in with observations which are far below the mean.
- The precision of the estimate is not a function of the size of the population, but of the size of the sample. A given size of sample will give the same precision whether we are sampling Ingham county or China (well, it’s a little more complex than that, but the point is generally correct).

- INSERT FIGURE 24 HERE.

- We are interested in whether a new IT system in a plant has accelerated the improvement in productivity in a plant. Suppose that over a long period we have found that productivity improved at an annualized rates of 3.0% per month with annualized variance of 1.5%. We look at the twelve months following installation of the new IT and find a growth rate of 3.5% annually. How likely are we to get this higher rate of growth if the plant is really on a 3% growth path. Growth rates are normally distributed.

- Issue: Because of sampling variability we know that our point estimate is unlikely to be exact. We would like to develop an estimator which allows for sampling variability.
- The problem with estimation: Shooting in a gallery in which we never see if we hit the target. Need to establish the properties of estimators theoretically

- Three properties which we like in an estimator:
- unbiased: the estimator does not systematically over or undershoot the population parameter of interest systematically.
- consistent: as the number of observations increases, the variability of our estimator decreases.
- Example: the variance of the sample mean falls at rate n

- efficient: for a given sample size, we prefer that our estimator have the smallest possible variance.

- Unbiased: The sampling distribution is centered around the population mean.
- A biased and unbiased estimator: We are measuring the number of units which flow by a point in a 10 second period. We have calibrated one instrument correctly. Unfortunately, the second instrument is mis-calibrated, and adds 1 to each count. The number of units which flow by the point is random but, as a sample, is normally distributed. We get the following chart.

- Need to be a little careful about using the term bias
- Example: For whatever reasons, we take a sample of men to estimate their absence behavior. This sample is likely a biased estimator of the absence behavior of the full population because male and female absence behavior is likely very different. It is an unbiased sample if we only use it to learn about male behavior. Whether this sample is biased or not depends on the population for which we use it. In the second instance, we acknowledge the limits of the sample (and so limit its usefulness) but the sample will be accurate (unbiased).

- In the instance of the male sample, we at least know that it is a biased estimator of overall absence behavior, even if we don’t know the direction of the bias. All to often, researchers don’t realize that they are using a biased estimator.
- Example: Send a survey out to 3,000 trucking firms, get 177 responses back

- Consistent: As the number of observations increases, our estimator becomes more precise.
- We know that the mean is a consistent estimator, is the median a consistent estimator?