Chapter 1: Sampling Distribution
Summary & Key concepts
Parameter: Population statistic
Sample statistic: A number describing a characteristic of a sample
Sampling space: All possible sample statistic values
Sampling distribution: All possible sampling statistic values and their probabilities or probability
densities
Probability density: A means of getting the probability that a continuous random variable (like
a sample statistic) falls within a particular range
Random variable: A variable with values that depend on chance
Expected value/ expectation: The mean of a probability distribution (= the true population
value)
Unbiased estimator: A sample statistic for which the expected value equals the population
value
Sampling distribution: Shows us the probability of drawing a sample with a particular sample
statistic
Sampling distributions are the central element in estimation and null-hypothesis testing
1. Draw thousands of samples
2. Calculate the mean
3. And you have the true population value
Complications with sampling distributions
1. Must be a random sample
2. The sample statistic must be an unbiased estimator of the population
3. Continuous versus discrete sample statistic: probability density versus probabilities
4. Impractical! (difficult to draw more than one sample)
1.1 Statistical inference: Making the most of your data
Inferential Statistics: Offers techniques for making statements about a larger set of
observations from data collected for a smaller set of observations
Allows you to generalize from a statement about the sample to a statement about the
population from which it was drawn
1.2 A Discrete Random Variable: How many yellow candies in my bag?
Sample statistic: A value describing a characteristic of the sample
Ex: The number of yellow candies in a bag
Sample statistics are random variables (the score depends on chance, the chance that a
particular sample is drawn)
1
,Sampling Distribution Example: the distribution of outcome stores of many different samples
Horizontal Axis = Sampling space (all the possible values that the sample statistic can have)
Left Hand Vertical Axis (count) = The number of samples that have been drawn with a particular
value for the sample statistic
Right Hand Vertical Axis = Proportions of previously drawn samples with a particular sample
statistic (also the probability that a previously drawn sample contains a particular number of
yellow candies)
Units of Analysis: In the left graph, the candies are the units of analysis but in the right graph (the
sampling distribution) samples (candy bags) are the units of analysis
Final Shape?: Theoretically the sampling distribution has its final shape after 1,000 or 2,000
samples but this is not always the case
Probability distribution of the sample statistic: A sampling space with a probability (between 0
and 1) for each outcome of the sample statistic
Discrete probability distribution: Probability distribution when there are a limited number of
outcome scores (ex: 0-10), it is possible to list all outcomes
Probabilities can be expressed as either a proportion (= number from 0-1) or percental
(0%-100%)
Expected value/ expectation of a probability distribution
Expected value = Mean of the sampling distribution
Changes in accordance with changes in the population proportion
Expected value = True population value (only if the sample statistic is an unbiased
estimate of the population value)
2
, If the proportion of yellow candies in the population is 0.2 (2/10) then the expected value
in a sample of 10 candies is 2 (0.2 (10)= 2)
Unbiased Estimator
A sample statistic is an unbiased estimator of the population statistic (parameter) if the
expected value (mean of the sampling distribution) is equal to the population statistic
This is often not the case because the population is much larger than the sample
(downward bias = underestimating the number in the population)
However, the proportion in the sample is an unbiased estimator of the population
proportion
Representative Sample
A sample is representative of a population if variables in the sample are distributed in the
same way as in the population
Random samples often differ from the population due to chance so although it is usually
not representative, we can say that it is representative in principle
Then we can use probability theory (construct confidence intervals) to account for
misrepresentation in the sample that we have drawn
3
, 1.3 Continuous Random Variable: Overweight and Underweight
Continuous Variables: Have an infinite number of values between any two values
Ex: Weight of a candy
Sample statistic (= weight of candy bag) we are more interested in the weight of a candy
bag than a single candy
Continuous Probabilities
The probability of getting a continuous sample statistic is basically zero (ex: you are very
unlikely to draw a sample bag with an average weight of exactly 2.8 grams)
Therefore, the probability distribution of the sampling space is not interesting (nearly all
zeros and impossible to list all the possible outcomes within the sampling space)
Probability Density
Looking at a range of values instead of a single value
Ex: The probability of having a sample bag with an average candy weight of between 2.75
and 2.85 grams
Ex 2: The probability of average candy weight above or below 2.8 grams
Probability Density Function: Gives us the probability of values between two thresholds
Used to display the probability of continuous random variables
Probability Density = area between the horizontal axis and the curve (probability density
function)
The probability of buying a bag with an average candy weight of 2.8 grams or more is 0.5
(right-hand probability)
Left Hand Probabilities: The probability of values up to (and including) a threshold value
Right Hand Probabilities: The probability of values above (and including) a threshold value
4
Summary & Key concepts
Parameter: Population statistic
Sample statistic: A number describing a characteristic of a sample
Sampling space: All possible sample statistic values
Sampling distribution: All possible sampling statistic values and their probabilities or probability
densities
Probability density: A means of getting the probability that a continuous random variable (like
a sample statistic) falls within a particular range
Random variable: A variable with values that depend on chance
Expected value/ expectation: The mean of a probability distribution (= the true population
value)
Unbiased estimator: A sample statistic for which the expected value equals the population
value
Sampling distribution: Shows us the probability of drawing a sample with a particular sample
statistic
Sampling distributions are the central element in estimation and null-hypothesis testing
1. Draw thousands of samples
2. Calculate the mean
3. And you have the true population value
Complications with sampling distributions
1. Must be a random sample
2. The sample statistic must be an unbiased estimator of the population
3. Continuous versus discrete sample statistic: probability density versus probabilities
4. Impractical! (difficult to draw more than one sample)
1.1 Statistical inference: Making the most of your data
Inferential Statistics: Offers techniques for making statements about a larger set of
observations from data collected for a smaller set of observations
Allows you to generalize from a statement about the sample to a statement about the
population from which it was drawn
1.2 A Discrete Random Variable: How many yellow candies in my bag?
Sample statistic: A value describing a characteristic of the sample
Ex: The number of yellow candies in a bag
Sample statistics are random variables (the score depends on chance, the chance that a
particular sample is drawn)
1
,Sampling Distribution Example: the distribution of outcome stores of many different samples
Horizontal Axis = Sampling space (all the possible values that the sample statistic can have)
Left Hand Vertical Axis (count) = The number of samples that have been drawn with a particular
value for the sample statistic
Right Hand Vertical Axis = Proportions of previously drawn samples with a particular sample
statistic (also the probability that a previously drawn sample contains a particular number of
yellow candies)
Units of Analysis: In the left graph, the candies are the units of analysis but in the right graph (the
sampling distribution) samples (candy bags) are the units of analysis
Final Shape?: Theoretically the sampling distribution has its final shape after 1,000 or 2,000
samples but this is not always the case
Probability distribution of the sample statistic: A sampling space with a probability (between 0
and 1) for each outcome of the sample statistic
Discrete probability distribution: Probability distribution when there are a limited number of
outcome scores (ex: 0-10), it is possible to list all outcomes
Probabilities can be expressed as either a proportion (= number from 0-1) or percental
(0%-100%)
Expected value/ expectation of a probability distribution
Expected value = Mean of the sampling distribution
Changes in accordance with changes in the population proportion
Expected value = True population value (only if the sample statistic is an unbiased
estimate of the population value)
2
, If the proportion of yellow candies in the population is 0.2 (2/10) then the expected value
in a sample of 10 candies is 2 (0.2 (10)= 2)
Unbiased Estimator
A sample statistic is an unbiased estimator of the population statistic (parameter) if the
expected value (mean of the sampling distribution) is equal to the population statistic
This is often not the case because the population is much larger than the sample
(downward bias = underestimating the number in the population)
However, the proportion in the sample is an unbiased estimator of the population
proportion
Representative Sample
A sample is representative of a population if variables in the sample are distributed in the
same way as in the population
Random samples often differ from the population due to chance so although it is usually
not representative, we can say that it is representative in principle
Then we can use probability theory (construct confidence intervals) to account for
misrepresentation in the sample that we have drawn
3
, 1.3 Continuous Random Variable: Overweight and Underweight
Continuous Variables: Have an infinite number of values between any two values
Ex: Weight of a candy
Sample statistic (= weight of candy bag) we are more interested in the weight of a candy
bag than a single candy
Continuous Probabilities
The probability of getting a continuous sample statistic is basically zero (ex: you are very
unlikely to draw a sample bag with an average weight of exactly 2.8 grams)
Therefore, the probability distribution of the sampling space is not interesting (nearly all
zeros and impossible to list all the possible outcomes within the sampling space)
Probability Density
Looking at a range of values instead of a single value
Ex: The probability of having a sample bag with an average candy weight of between 2.75
and 2.85 grams
Ex 2: The probability of average candy weight above or below 2.8 grams
Probability Density Function: Gives us the probability of values between two thresholds
Used to display the probability of continuous random variables
Probability Density = area between the horizontal axis and the curve (probability density
function)
The probability of buying a bag with an average candy weight of 2.8 grams or more is 0.5
(right-hand probability)
Left Hand Probabilities: The probability of values up to (and including) a threshold value
Right Hand Probabilities: The probability of values above (and including) a threshold value
4
