Statistics: Variance, Standard Deviation and Sampling
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Questions and Answers

Which of the following is the correct formula for calculating the variance of a dataset?

  • $\sqrt{\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2}$
  • $\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})^2$ (correct)
  • $\frac{1}{n} \sum_{i=1}^{n} (x_i - \bar{x})$
  • $\sum_{i=1}^{n} (x_i - \bar{x})^2$

What does the standard deviation represent?

  • The difference between the maximum and minimum values in a dataset.
  • The square root of the variance. (correct)
  • The square of the variance.
  • The average deviation from the mean.

When is the coefficient of variation (CV) most useful?

  • When comparing the variability of datasets with different units or means. (correct)
  • When comparing the means of two datasets.
  • When the datasets have the same mean.
  • When the standard deviation is zero.

If a dataset has a mean of 50 and a standard deviation of 10, what is the coefficient of variation?

<p>0.2 (C)</p> Signup and view all the answers

What does the 0.5 quantile (Q(0.5)) represent?

<p>The value below which 50% of the observations fall. (D)</p> Signup and view all the answers

Which of the following is another name for the 0.5 quantile?

<p>Median (A)</p> Signup and view all the answers

Which of these represent quartiles?

<p>Q(0.25), Q(0.5), Q(0.75) (C)</p> Signup and view all the answers

What values do deciles divide a dataset into?

<p>10 equal parts (C)</p> Signup and view all the answers

A researcher aims to study the average income of all software engineers in Europe. What constitutes the 'population' in this scenario?

<p>All software engineers in Europe, regardless of their employment status or location. (C)</p> Signup and view all the answers

Which of the following best describes 'sampling bias'?

<p>A systematic error in the sampling process that leads to a non-representative sample. (C)</p> Signup and view all the answers

In the context of statistical inference, what does it mean to 'infer'?

<p>To deduce or conclude information from evidence and reasoning. (D)</p> Signup and view all the answers

A polling company only surveys individuals who own smartphones to gauge public opinion on a new technology. What type of bias is most likely to affect the results of this survey?

<p>Sampling bias (D)</p> Signup and view all the answers

What is the primary purpose of random sampling?

<p>To give each member of the population an equal chance of being selected, reducing bias. (D)</p> Signup and view all the answers

A researcher is studying the job satisfaction of employees at a large corporation. They distribute surveys only to employees in the marketing department. What is the most significant concern regarding this sampling method?

<p>The sample may not be representative of the entire corporation, leading to sampling bias. (A)</p> Signup and view all the answers

In the context of sampling, what is 'sampling error'?

<p>The difference between a sample statistic and the true population parameter due to chance. (C)</p> Signup and view all the answers

What primarily causes the difference between a sample statistic and a population parameter?

<p>Sampling error arising by chance. (B)</p> Signup and view all the answers

A university wants to assess student satisfaction with their academic programs. Which sampling method would be LEAST likely to introduce sampling bias?

<p>Randomly selecting students from a comprehensive list of all enrolled students. (D)</p> Signup and view all the answers

In the context of sampling, what does the term 'N' typically represent?

<p>The size of the population. (C)</p> Signup and view all the answers

If $\mu_x$ represents the population mean and $\bar{x}$ represents the sample mean, what is the expected relationship between $\mu_x$, $\bar{x}$, and the sample size n as n increases?

<p>$\bar{x}$ will converge towards $\mu_x$ as <em>n</em> increases, due to the law of large numbers. (B)</p> Signup and view all the answers

What is the likely effect of taking many random samples from a population and calculating the mean of each sample?

<p>The sample means will vary, showing a distribution around the population mean. (D)</p> Signup and view all the answers

What does the height of a bar in a histogram typically represent?

<p>The fraction of observations that fall within that bin, representing relative frequency. (B)</p> Signup and view all the answers

Suppose you are analyzing income data from Finland in 2010. If you increase your sample size from n = 100 to n = 1000, what would you expect to observe regarding the distribution of sample means?

<p>The distribution of sample means will become narrower (less spread out). (B)</p> Signup and view all the answers

In the context of a histogram, what is the primary purpose of dividing observations into bins?

<p>To group continuous data into manageable intervals for frequency distribution analysis. (A)</p> Signup and view all the answers

Which of the following actions would likely NOT reduce sampling error when estimating a population parameter?

<p>Using a biased sampling method. (B)</p> Signup and view all the answers

Which statement accurately describes how observations are allocated to bins in a typical histogram?

<p>Each observation is allocated to a single bin based on its value falling within the bin's range. (C)</p> Signup and view all the answers

Imagine two researchers are studying the average height of adults in a city. Researcher A takes a sample of 50 people, while Researcher B takes a sample of 500 people. Assuming both researchers use the same random sampling method, which researcher is likely to have a sample mean that is closer to the true population mean, and why?

<p>Researcher B, because larger samples reduce sampling error. (B)</p> Signup and view all the answers

If you increase the number of bins in a histogram for a fixed dataset, what is the likely effect on the appearance of the histogram?

<p>The histogram will show a more granular view of the data distribution, potentially revealing finer patterns. (A)</p> Signup and view all the answers

A researcher calculates a sample mean income of $30,000 from a random sample. The population mean income is $32,000. Which statement best describes this situation?

<p>The $2,000 difference might be due to sampling error. (B)</p> Signup and view all the answers

What does the width of a bin in a histogram represent?

<p>The range of values that observations within the bin can take. (B)</p> Signup and view all the answers

A histogram is best suited for visualizing the distribution of what type of variable?

<p>Continuous or discrete variables, showing the frequency of values within specified intervals. (A)</p> Signup and view all the answers

Which of the following is NOT a typical characteristic of a histogram?

<p>Gaps between bars to indicate separation between categories. (B)</p> Signup and view all the answers

If a histogram is described as skewed to the right, what does this indicate about the underlying data distribution?

<p>The data has a long tail extending towards higher values. (D)</p> Signup and view all the answers

Based on the kernel density estimate provided, what would be the best method to estimate the fraction of samples with incomes above $40,000?

<p>Calculate the area under the curve to the <em>left</em> of $40,000, then subtract this from 1. (B)</p> Signup and view all the answers

Given a cumulative density function (CDF) $F_X(t)$, how do you interpret the value of $F_X(50)$?

<p>The proportion of observations with values less than or equal to 50. (B)</p> Signup and view all the answers

What does the bandwidth parameter in a kernel density estimator primarily control?

<p>The smoothness of the resulting density estimate. (B)</p> Signup and view all the answers

Why is it important to choose an appropriate bandwidth when using a kernel density estimator?

<p>To avoid distorting the underlying distribution of the data. (A)</p> Signup and view all the answers

What is the primary difference between a probability density function (PDF) and a cumulative density function (CDF)?

<p>A PDF gives the density at a point; a CDF gives the probability of being less than or equal to a point. (A)</p> Signup and view all the answers

If the CDF, $F_X(x)$, of a random variable $X$ is given by $F_X(x) = 1 - e^{-2x}$ for $x \geq 0$, what is the probability that $X$ is greater than 1?

<p>$e^{-2}$ (D)</p> Signup and view all the answers

For the kernel density estimate shown in the image where kernel = epanechnikov, what effect would decreasing the bandwidth from 2.7e+03 have on the resulting density curve?

<p>The density curve would become more jagged and potentially show more local features. (B)</p> Signup and view all the answers

A researcher wants to compare the income distributions of two different cities using kernel density estimates. They use the same bandwidth for both cities. What potential problem might arise from this approach?

<p>The optimal bandwidth might be different for each city, leading to misleading comparisons. (A)</p> Signup and view all the answers

How does increasing the sample size affect the distribution of sample averages in relation to the population mean?

<p>Larger sample sizes cause sample averages to cluster more closely around the population mean. (B)</p> Signup and view all the answers

What is the general shape of the distribution of sample averages around the population mean, based on repeated random sampling?

<p>Distributed relatively symmetrically. (A)</p> Signup and view all the answers

A researcher is studying income levels in a city. They take multiple random samples of different sizes and calculate the average income for each sample. Which sample size is most likely to provide an average income closest to the true average income of the entire city?

<p>A sample of 5,000 households. (D)</p> Signup and view all the answers

A statistical analysis produces several sample averages from different sample sizes. Which of the following statements accurately describes the expected relationship between sample size and the proximity of the sample average to the population mean?

<p>Larger samples are more likely to have averages closer to the population mean due to the law of large numbers. (A)</p> Signup and view all the answers

Suppose a researcher collects multiple random samples to estimate the average height of adults in a city. Which situation would result in the most reliable estimate of the population mean?

<p>Large samples from diverse neighborhoods across the city. (B)</p> Signup and view all the answers

If you repeatedly draw random samples from a population and calculate a statistic (e.g., mean, standard deviation) for each sample, the distribution of these statistics is called the:

<p>Sampling distribution (D)</p> Signup and view all the answers

What does a narrower sampling distribution of the mean indicate?

<p>The sample means are more consistent and closer to the population mean. (B)</p> Signup and view all the answers

A researcher wants to estimate the average lifespan of a particular species of insect. To improve the accuracy and reliability of their estimate, which action should they prioritize when collecting samples?

<p>Increase the size of each sample and ensure random selection. (B)</p> Signup and view all the answers

Flashcards

Variance

A measure of how spread out numbers are in a dataset.

Standard Deviation

The square root of the variance; measures the spread of data around the mean.

Coefficient of Variation (CV)

Standard deviation divided by the mean; useful for comparing variability across different scales.

Quantile Q(p)

The value below which a fraction 'p' of the data falls.

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Median [Q(.5)]

The point separating the higher half from the lower half of a data sample, a population, or a probability distribution.

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Quartiles: Q(.25), Q(.5), Q(.75)

Values that divide the data into four equal parts.

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Deciles: Q(.1), Q(.2),..., Q(.9)

Values that divide the data into ten equal parts.

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Percentiles: Q(.01), Q(.02),..., Q(.99)

Values that divide the data into one hundred equal parts.

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What is a histogram?

The empirical counterpart of the density function for a discrete variable.

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What does the height of a histogram bar represent?

The fraction of observations that fall into a given category or bin.

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What is a bin in a histogram?

A range or interval of values into which observations are grouped.

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Observation Allocation

Each data point is assigned exclusively to one bin.

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Complete Bin Allocation

All data points are contained within at least one bin.

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What defines the bin width?

The range of values that observations within that bin can take.

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What is the general purpose of observations divided into bins?

Dividing observations into intervals to represent data distribution.

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What is the main use of a histogram?

To visualize the distribution of a variable based on observed data.

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Population

The entire group that you want to draw conclusions about.

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Sample

A specific group selected from the population to collect data.

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Infer

To deduce or conclude information from evidence and reasoning, rather than explicit statements.

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Sampling Bias

When the sample does not accurately represent the population.

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Sampling Error

When exceptional observations are sampled by chance, leading to unrepresentative results.

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Straw Poll

Polling where individuals self-select to participate, which can skew results.

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Random Sampling

Selecting a sample such that each object in the population has an equal chance of being chosen.

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Random Sampling: Reduces Bias

A method to mitigate sampling bias by ensuring every member of the population has an equal chance of selection.

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Density Plot

A graphical representation showing the distribution of a continuous variable.

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Kernel Density Estimator

A non-parametric way to estimate the probability density function of a random variable.

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Bandwidth

Controls the smoothness of the density estimate. A smaller bandwidth creates a more detailed, less smooth curve; a larger bandwidth creates a smoother curve.

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Kernel Function

A function that determines the shape of the curve used to estimate the density at each data point. Epanechnikov is one common type of kernel.

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Cumulative Density Function (CDF)

A function that gives the probability that a random variable X is less than or equal to a certain value t.

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CDF Formula

FX(t) = P(X ≤ t); the probability that the random variable X takes on a value less than or equal to t.

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CDF and Area

It's the area under the probability density function (PDF) up to a certain value.

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CDF Purpose

Answers the question: What proportion of observations are below a certain value?

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Effect of Sample Size

The tendency of sample averages to be closer to the population mean when the sample size is larger.

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Distribution of Sample Averages

Sample averages tend to be distributed symmetrically around the true population mean.

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Sample Average

The average value calculated from a sample.

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Population Mean

The average value of all data points in a population.

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Statistical Inference

Using samples to estimate parameters of a population.

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Population Mean (µx)

The true average value of a characteristic in the entire population.

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Sample Mean (x̄)

The average value calculated from a subset (sample) of the population.

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Sample Size (n)

The number of individuals or observations included in a sample.

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Population Size (N)

The total number of individuals or observations in the entire group of interest.

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Impact of Sample Size

The difference between the sample mean and the population mean decreases as the sample size increases.

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Random Sample

Selecting a subset of individuals from within a statistical population to estimate characteristics of the whole population

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Population Mean Example

Income average if we surveyed all 15–64 year olds in Finland in 2010.

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Study Notes

  • Principles of Empirical Analysis (ECON-A3000) Lecture 2 is about samples and descriptive statistics

Logistics

  • Bring name placards to class
  • Pre-class assignment 1 was due 15 minutes before class.
  • Up to two skips are allowed without penalty
  • Grade is pass/fail based on effort
  • There is an in-class worksheet due at the end of class
  • The worksheet should be picked up from upfront.
  • A photo or scan of the worksheet should be submitted to MyCourses before the next class
  • The in-class worksheet will be pass/fail based on accuracy.

Learning Objectives

  • The learning objectives for the lecture include:
    • Descriptive statistics (mean, variance, standard deviation, median and quantiles, density functions, joint distributions)
    • Sample and population (representativeness, sampling error)

Descriptive Statistics

  • Descriptive statistics are ways of summarizing information to make data understandable.
  • The objective is to reduce the amount of numbers which losing as little information as possible.
  • Stata's summarize command gives the key descriptive statistics, including:
    • sample mean
    • a single number measures of variation
    • selected quantiles

Measures of Variation

  • Variance formula: Var(x) = 1/n * Σ (xi - xÌ„)²
  • Standard deviation formula: SD(x) = √Var(x)
  • The coefficient of variation allows comparison across variables by normalizing the standard deviation with the mean
  • Coefficient of variation formula: CV(x) = SD(x) / xÌ„

Quantiles

  • Definition: Quantile Q(p) is the value such that a fraction p of observations take at most value Q(p)
  • Some quantiles have specific names. e.g. median.
  • Q(0.5) indicates that 50% of the observations are below this value.
  • Some other named quantiles include: quartiles, deciles, and percentiles.
  • Quartiles: Q(0.25), Q(0.5), Q(0.75)
  • Deciles: Q(0.1), Q(0.2), ..., Q(0.9)
  • Percentiles: Q(0.01), Q(0.02), ..., Q(0.99).
  • Distribution width is characterized with percentile ratios
  • 90/10 ratio = Q(0.9)/Q(0.1) = 15
  • 90/50 ratio = Q(0.9)/Q(0.5) = 2.1
  • 50/10 ratio = Q(0.5)/Q(0.1) = 7

Density functions

  • For a discrete random variable X, the density function is fX(x) = P(X = x), representing the probability that X takes a specific value x.
  • For density functions must hold the following conditions: fX(x) ≥ 0 and Σ fX(x) = 1.
  • The probability that X takes a value within set A is P(X∈ A) = Σ fX(x) for all x in A.
  • A histogram is the empirical counterpart of the density function for a discrete variable.
  • The bar height describes the fraction of observations that take the x value.
  • Bins are used to divide the data into separate groups to draw a histogram.
  • Each observation is allocated to a single bin, and all observations are allocated to some bin.
  • The width of the bin describes the values that observations within the bin can take.
  • Changing the number of bins may change how the data is viewed.
  • If X is a continuous random variable, the probability that X takes a value within the set A is: P(X ∈ A) = ∫ fX(x)dx over A.
  • Continuous variables can take infinite values giving a 0 value, i.e., P(X = x) = ∫ fX(x)dx = 0.
  • Density function can be interpreted w.r.t. to small variation, h > 0,
  • Approximate Formula: fX(x) ≈ P(X = x ± h/2) / h,
  • The definition of a kernel uses this formula.
  • A kernel density estimator is a local weighted average for each value x.
  • The formula for kernel density estimators is: fÌ‚h(x) = 1/n * Σ Kh(x - xi), where the sum is from i = 1 to n.
  • Bandwidth(h) measures the amount of data around x.
  • Kernel function(Kh) measures how to weight observations.
  • By default, Stata chooses an optimal bandwidth.
  • Larger bandwidth disregards more data and smaller bandwidth creates more noise.
  • The Cumulative Density Function (CDF) for a continuous variable: Fx(t) = ∫ fX(s)ds.
  • A CDF answers what fraction of the observations have values of x below t
  • For a standardized normal distribution Fx(-1) = 0.159, and Fx(0) = 0.5

Population and Sample

  • A population is all units one wants to draw conclusions about (N units).
  • A sample is a specific group selected from the population to collect data (n units).
  • The goal is to make an inference of the larger population.
  • Inference definition: Deduce or conclude (information) from evidence and reasoning rather than from explicit statements.
  • Sampling bias occurs when the sample is not representative of the population.
  • Sampling error occurs when exceptional observations are sampled by chance.
  • Random sampling removes bias.
  • In random sampling each object has the same probability of being selected into the sample.
  • Sampling error remains, because of the difference between a sample statistic and the overall population parameter.

Sampling Error: Example

  • The population mean income among 15-64 year olds living in Finland in 2010 is 26,144 euros (N ≈ 3.5M).
  • Using a random sample of n people to calculate the sample average, x-bar = 1/n * Σ xi
  • The larger the sample size, the closer the sample average will be to the population mean.
  • Sample averages are distributed relatively symmetrically around the population mean
  • These properties are also known as:
    • The Law of Large Numbers
    • The Central Limit Theorem
  • These facts are deep results discussed more formally in later econometrics courses.

Joint Distributions

  • Cross tabulation is an efficient way to display (small) data for two variables
  1. Rows are the number of values that Y can take
  2. Columns are the number of values that X can take
  3. Cells report the number of observations with value (y, x)
  • Cross tabulation cells can report the share of observations as well.
  • The empirical counterpart of the joint density function: fxy(x, y) = P(X = x, Y = y)
  • i.e., the probability that X takes the value x and Y takes the value y.

Summary

  • Covered concepts for understanding:
    • Density function, CDF
    • Joint distributions
  • Considerations when using samples
    • Representativeness
    • Sampling error

Assignments

  • In-class worksheet 1 is due on MyCourses before the next lecture
  • Submit a preferably a photo/scan, or turn in a paper copy at the beginning of the next lecture
  • Pre-class assignment 2 is due 15 minutes before the next lecture
  • Homework 1 is due on Jan 15
  • Now the conceptual tools to are known to get started
  • It is a good idea to attend Exercise Session 1 tomorrow for practical tools.

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Test your knowledge of statistics, including variance and standard deviation. Explore quantiles, quartiles, and deciles. Also, understand the concepts of population, sampling bias, statistical inference, and random sampling.

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