Chapter 1: Sampling Distribution
1.1 Statistical inference
● Statistics confirm if theories are true through data collection
● Data collection= expensive and time consuming→ aim is to collect as little data as
possible while describing a large data set.
● Inferential statistics: making statements about a population based on data collected from
small amount of observation (sample); generalizing a statement about a sample to a
statement about a population
● Population: a large set of observations
● Sample: a small set of observations
● Random sampling is the most common sampling method in this field
● Statistical inference: generalizing data collected from a random sample of a specific
population and applying it to the population.
1.2 A discrete random variable: how many yellow candies in the bag?
1.2.1 Sample statistic
● Sample statistic: the number associated with the characteristic of the population we are
interested in
● Sampling space: all possible outcome scores in a sample
○ Random variable: a variable that changes depending on the sample
1.2.2 Sampling distribution
● Sampling distribution: layout of outcome scores from many samples
● Histogram horizontal axis represents sample space
● Histogram vertical axis represents sample count (how many time that sample space was
taken)
● In the yellow candy example, there are 2 units of analysis in histogram (candy color and
number of samples taken)
1.2.3 Probability and probability distribution
● Probability distribution: sampling space with probability between 0 and 1 for each
outcome of the sample statistic
● Discrete probability: when ther ei s a limited number of outcomes possible
● If the population proportion is 0.5, the probability distribution is symmetric
● Sampling distributions tell us what outcomes to expect
● Discrete probability distributions: limited number of outcomes possible and you can find
out the probability of each outcome separately
1.2.4 Expected value or expectation
1
, ● Mean of sampling distribution is the expected value o f sample statistic
● Mean of sampling distribution is equal to the population proportion/ population value (if
unbiased estimator
● Mean represents the balance point of a distribution (symmetrical)
● Expected value: average of the sampling distribution of a random variable
1.2.5 Unbiased estimator
● Population statistic is also known as the parameter
● Most sample statistics are unbiased estimators of population statistics
● Downward biased: saying that there is 2 yellow candies in the parameter bc there were 2
in our sample of 10→ too low
● Unbiased estimator comes from our sample/sample distribution→ you cannot use sample
statistic to generalize from sample to population
1.2.6 Representative sample
● We expect the shares of each sample statistic in our sample to be representative and equal
to their shares in the population.
● A sample is representative is variables are distributed the same way in a population as it
is in the sample.
● A random sample can never be representative because of chance but we should expect it
to be representative so we say that it is in principle representative of the population.
● The probability theory accounts for the misrepresentation in the actual sample we draw→
done through statistical inference to make confidence intervals and null hypothesis
testing.
1.3 A continuous random variable: overweight and underweight
1.3.1 Continuous variable
● Continuous variable: a variable that has a new value between two values.
1.3.2 Continuous sample statistic
● Sample statistic= average candy weight in sample (so probability we will extract a
sample with a certain weight)
● Sample mean is an unbiased estimator of the population mean
○ Avg candy weight in population (at factory) is the avg of the candy weight in
sampling distribution.
○ Also the avg weight we expect to extract from population
1.3.3 Continuous probabilities
● Almost impossible to extract a specific candy weight (ex. 2.8) bc we can’t exclude 2.81
or 2.801 etc
● Probability of extracting a specific fixed candy weight is zero
● Probability distribution will be all zeros and boring (continuous distribution= threshold)
1.3.4 p Values
● To solve this problem, we look at a range of values
2
, ● Choose a threshold and use it to find the probability of extracting sample with means
above and below it
● Probability is displayed as an area between the horizontal axis and curve
● Curve= probability density function (x axis is labelled probability density-- values placed
on x axis); area is always equal to 1
● P-value: the probability of values up to and including the threshold value or the threshold
value and higher
○ Left hand p-value: values including threshold value and lower
○ Right hand p-value: values including threshold values and higher
● Histograms?
○ Probability sampling distribution (discrete) vs probability density distribution
(continuous)
○ Histogram gives you a range for the continuous variables
○
1.3.5 probabilities always sum to 1
● Can find threshold values knowing probabilities by subtracting probabilities from 1
1.4 Concluding remarks
1.4.1 Samples characteristic as observation
● Each sample is a case which adds an observation (sample characteristic) to the sampling
distribution
● Its sample statistic value is what is added to the sampling distribution (sample
characteristic)
1.4.2 Means at three levels
● Sample mean= means at population level, mean at
sampling distribution level (“the mean of means”),
mean at sample level (average of candy weights in
sample we extract)
● Population mean is same value as sampling
distribution expected value
○ Expected value of sampling distribution is
average of sampling distribution
● Sample mean could be different than population
mean and sampling distribution mean because the
latter two are the same and the sample mean is drawn
at random so it can always change and be different.
○ Sample mean is more likely to be close to the
population mean but it can still be different
3
, 1.5 Test your understanding
1) In the real world population distribution, the population mean would be 2 because the
sample distribution is an unbiased estimator of the population distribution
2) The number on the horizontal axis represent the number of yellow candies in the sample
drawn. The statistical name of the variable is the sample statistic. The unit of analysis for
this characteristic is candies and candy color.
3) The sample characteristic values range from 1 to 10. This set of values is called the
sample space.
4) The number of yellow candies per sample that appears most frequently is 2 because that
is the proportion of yellow candies in the population.
5) The color distribution in each sample is not representative of the color distribution in the
stock of candies but we say that in principle it is.
6) The sample characteristic (sample statistic) is called a random variable because it can
change each time we draw a sample from the population.
7) The sampling distribution represent the probability density function of the average
sample candy weights drawn. The area under the curve represents the probability to draw
a sample with a certain range of candy weights (threshold values)
8) The average weight of all candies in the population is 2.8g. We know this because the
expected value of the sampling distribution is the expected value of the population
distribution when the sample is an unbiased estimator of the population.
9) Probability of drawing sample with avg candy weight between 2 and 2.9g is 0.479
10) The probability of drawing a sample with average candy weight of 2.9g exactly is zero
because it is impossible to draw one exactly at that weight or at least the probability is
very small.
11) This graph is an example of a continuous probability distribution because there is an
unlimited amount of values possible to be drawn and there can always be a new value
drawn between two values.
12) The vertical axis is labelled probability density instead of probability because in a
probability density function, the area under the curve is used to measure the probability
of drawing a sample with a certain value range.
The difference between population distribution, sampling distribution, and sample
distribution?
● Population distribution: the entire population→ from that you take a sample
● Sample distribution: how many different characteristics are in ONE SAMPLE
● Sampling distribution: shows the number of times the sample statistic shows up across
ALL SAMPLES
Why is sampling distribution important?
4
1.1 Statistical inference
● Statistics confirm if theories are true through data collection
● Data collection= expensive and time consuming→ aim is to collect as little data as
possible while describing a large data set.
● Inferential statistics: making statements about a population based on data collected from
small amount of observation (sample); generalizing a statement about a sample to a
statement about a population
● Population: a large set of observations
● Sample: a small set of observations
● Random sampling is the most common sampling method in this field
● Statistical inference: generalizing data collected from a random sample of a specific
population and applying it to the population.
1.2 A discrete random variable: how many yellow candies in the bag?
1.2.1 Sample statistic
● Sample statistic: the number associated with the characteristic of the population we are
interested in
● Sampling space: all possible outcome scores in a sample
○ Random variable: a variable that changes depending on the sample
1.2.2 Sampling distribution
● Sampling distribution: layout of outcome scores from many samples
● Histogram horizontal axis represents sample space
● Histogram vertical axis represents sample count (how many time that sample space was
taken)
● In the yellow candy example, there are 2 units of analysis in histogram (candy color and
number of samples taken)
1.2.3 Probability and probability distribution
● Probability distribution: sampling space with probability between 0 and 1 for each
outcome of the sample statistic
● Discrete probability: when ther ei s a limited number of outcomes possible
● If the population proportion is 0.5, the probability distribution is symmetric
● Sampling distributions tell us what outcomes to expect
● Discrete probability distributions: limited number of outcomes possible and you can find
out the probability of each outcome separately
1.2.4 Expected value or expectation
1
, ● Mean of sampling distribution is the expected value o f sample statistic
● Mean of sampling distribution is equal to the population proportion/ population value (if
unbiased estimator
● Mean represents the balance point of a distribution (symmetrical)
● Expected value: average of the sampling distribution of a random variable
1.2.5 Unbiased estimator
● Population statistic is also known as the parameter
● Most sample statistics are unbiased estimators of population statistics
● Downward biased: saying that there is 2 yellow candies in the parameter bc there were 2
in our sample of 10→ too low
● Unbiased estimator comes from our sample/sample distribution→ you cannot use sample
statistic to generalize from sample to population
1.2.6 Representative sample
● We expect the shares of each sample statistic in our sample to be representative and equal
to their shares in the population.
● A sample is representative is variables are distributed the same way in a population as it
is in the sample.
● A random sample can never be representative because of chance but we should expect it
to be representative so we say that it is in principle representative of the population.
● The probability theory accounts for the misrepresentation in the actual sample we draw→
done through statistical inference to make confidence intervals and null hypothesis
testing.
1.3 A continuous random variable: overweight and underweight
1.3.1 Continuous variable
● Continuous variable: a variable that has a new value between two values.
1.3.2 Continuous sample statistic
● Sample statistic= average candy weight in sample (so probability we will extract a
sample with a certain weight)
● Sample mean is an unbiased estimator of the population mean
○ Avg candy weight in population (at factory) is the avg of the candy weight in
sampling distribution.
○ Also the avg weight we expect to extract from population
1.3.3 Continuous probabilities
● Almost impossible to extract a specific candy weight (ex. 2.8) bc we can’t exclude 2.81
or 2.801 etc
● Probability of extracting a specific fixed candy weight is zero
● Probability distribution will be all zeros and boring (continuous distribution= threshold)
1.3.4 p Values
● To solve this problem, we look at a range of values
2
, ● Choose a threshold and use it to find the probability of extracting sample with means
above and below it
● Probability is displayed as an area between the horizontal axis and curve
● Curve= probability density function (x axis is labelled probability density-- values placed
on x axis); area is always equal to 1
● P-value: the probability of values up to and including the threshold value or the threshold
value and higher
○ Left hand p-value: values including threshold value and lower
○ Right hand p-value: values including threshold values and higher
● Histograms?
○ Probability sampling distribution (discrete) vs probability density distribution
(continuous)
○ Histogram gives you a range for the continuous variables
○
1.3.5 probabilities always sum to 1
● Can find threshold values knowing probabilities by subtracting probabilities from 1
1.4 Concluding remarks
1.4.1 Samples characteristic as observation
● Each sample is a case which adds an observation (sample characteristic) to the sampling
distribution
● Its sample statistic value is what is added to the sampling distribution (sample
characteristic)
1.4.2 Means at three levels
● Sample mean= means at population level, mean at
sampling distribution level (“the mean of means”),
mean at sample level (average of candy weights in
sample we extract)
● Population mean is same value as sampling
distribution expected value
○ Expected value of sampling distribution is
average of sampling distribution
● Sample mean could be different than population
mean and sampling distribution mean because the
latter two are the same and the sample mean is drawn
at random so it can always change and be different.
○ Sample mean is more likely to be close to the
population mean but it can still be different
3
, 1.5 Test your understanding
1) In the real world population distribution, the population mean would be 2 because the
sample distribution is an unbiased estimator of the population distribution
2) The number on the horizontal axis represent the number of yellow candies in the sample
drawn. The statistical name of the variable is the sample statistic. The unit of analysis for
this characteristic is candies and candy color.
3) The sample characteristic values range from 1 to 10. This set of values is called the
sample space.
4) The number of yellow candies per sample that appears most frequently is 2 because that
is the proportion of yellow candies in the population.
5) The color distribution in each sample is not representative of the color distribution in the
stock of candies but we say that in principle it is.
6) The sample characteristic (sample statistic) is called a random variable because it can
change each time we draw a sample from the population.
7) The sampling distribution represent the probability density function of the average
sample candy weights drawn. The area under the curve represents the probability to draw
a sample with a certain range of candy weights (threshold values)
8) The average weight of all candies in the population is 2.8g. We know this because the
expected value of the sampling distribution is the expected value of the population
distribution when the sample is an unbiased estimator of the population.
9) Probability of drawing sample with avg candy weight between 2 and 2.9g is 0.479
10) The probability of drawing a sample with average candy weight of 2.9g exactly is zero
because it is impossible to draw one exactly at that weight or at least the probability is
very small.
11) This graph is an example of a continuous probability distribution because there is an
unlimited amount of values possible to be drawn and there can always be a new value
drawn between two values.
12) The vertical axis is labelled probability density instead of probability because in a
probability density function, the area under the curve is used to measure the probability
of drawing a sample with a certain value range.
The difference between population distribution, sampling distribution, and sample
distribution?
● Population distribution: the entire population→ from that you take a sample
● Sample distribution: how many different characteristics are in ONE SAMPLE
● Sampling distribution: shows the number of times the sample statistic shows up across
ALL SAMPLES
Why is sampling distribution important?
4
