Chapter 1: Sampling Distribution
Summary
- Statistical inference is about estimation and null hypothesis testing
- We collect data on a random sample and we want to draw conclusions (make inferences) about
the population from which the sample was drawn
- The sample does not offer a perfect image of the population (drawing another random sample
would have different characteristics)
- To make an informed decision on the CI we must compare the characteristics of the sample we
have drawn to the ones of the sample we could have drawn
- Sampling distribution = characteristics of the samples that we could have drawn
1.1 Statistical inference: making the most of your data
- Statistics is a tool for scientific research
- Collecting data is expensive
- Inferential statistics offers techniques for making statements about a larger set of observations
from data collected for a smaller set of observations
- The large set of observation about which we want to make a statement is called the population
- The smaller set is called a sample
- Statistical inference is a generalisation from the data collected in a random sample to the
population from which the sample was drawn
1.2 A discrete random variable: how many yellow candies in my bag?
Sample statistic
- We are usually interested in a particular characteristic of the sample rather than in the exact
nature of each observation within the sample
- Sample statistic: a number describing a characteristic of the sample
- Each sample has one outcome score on the sample statistic
- All possible outcome scores constitute the sampling space
- Sample statistic is a random variable
Sampling distribution
- Some sample statistic outcomes occur more often than other outcomes
- Sampling distribution = distribution of outcome scores of very many samples
Probability and probability distribution
- What is the probability of drawing a sample with five yellow candies as a sample statistic
outcome? This probability is the proportion of all possible samples that we could have drawn that
happen to contain five yellow candies
- Sampling distribution tells us all possible samples that we could have drawn
,- If we change the frequencies in the sampling distribution into proportions, we obtain the
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: because only a limited number of outcomes are possible
- Sampling distribution as a probabiloty distribution conveys very important information
• Tells us which outcomes we can expects and the probability that a particular outcome may
occur
Expected value or expectation
- If the share of yellow candies in the population is 0.20, we expect one out of each five candies in
a bag to be yellow
- The expected value of the proportion of yellow candies in the sample is equal to the proportion of
yellow candies in the population
- Expected value = the average of the sampling distributions of a random variable
- The sampling distribution is an example of a probabiloty distribution, the excepted value is the
average of a probability distribution
- The expected value is also called the expectation of a probability distribution
Unbiased estimator
- Expected value of the proportion of the sample statistic equal the true proportion of the
population statistic
- The sample proportion is an unbiased estimator of the proportion in the population
- Unbiased estimator = when expected value (mean of the sampling distribution) is equal to the
population statistic
- Parameter = population statistic
- Downward biased = it is too low (population much larger than the sample)
Representative sample
- A sample is representative of a population if variables in the sample are distributed in the same
way as in the population
- A random sample is likely to differ from the population due to chance, so the actual sample that
we have drawn is usually not representative of the population
- BUT we should expect it to be representative, so we say that it is in principle representative of
the population
1.3 A continuous random variable: overweight and underweight
Continuous variable
- Weight is a continuous variable because we can always think of a new weight between two other
weights
- If we can always think of a new value between two values, the variable is continuous
,Continuous sample statistic
- We are interested in the average weight of all candies in our sample bag, so average candy weight
in our sample is our key sample statistic
- The sample mean is an unbiased estimator of the population mean, so the average weight of all
candies in the population is the average of the sampling distribution
- And this is the average weight that we expect in a sample drawn from this population
Probability density
- We can solve this problem by looking at a range of values instead of a single value
- Choose a threshold and determine the probability of values above or below this threshold
- Probability density function = a curve, so if we cannot determine the probabiloty of a single
value, we display probabilities between the horizontal axis and this curve
- Probability density function = can gives us the probability of values between two thresholds
• It can also five us the probability of values up to (and including) a threshold value, which is
known as a left-hand probability or the probability of values above (and including) a
threshold value, which is called a right-hand probability
1.4 Concluding Remarks
Sample characteristics as observations
- Sampling distributions are our cases (units of analysis) and sample characteristics are our
observations
- In a sampling distribution, we observe samples (cases) and measure a sample statistic as the
(random) variable
- Each sample adds one observation to the sampling distribution and its sample statistic is the value
added to the sampling distribution
Means at three levels
- The mean of the sampling distribution is the average of the average weight of candies in every
possible sample bag
- This mean of means had the same value as our first mean, namely the average weight of the
candies in the population because a sample mean is an unbiased estimator of the population
mean
- The population and the sample consist of the same type of observations
Take-home points
- Values of a sample static vary across random samples from the sample populations. But some
values are more probable than other values
- The sampling distributions of a sample statistic tells us the probability of drawing a sample with a
particular value of the sample statistic or a particular min/max value
, - If a sample statistic is an unbiased estimator of a parameter, the parameter value is the average of
the sampling distribution, which is called the expected value or expectations
- For discrete sample statistics, the sampling distribution tells us the probability of individual
sample outcomes. For continuous sample statistics, it tells us the probability density, which gives
us the probability of drawing a sample with an outcome that is at least or at the most a particular
value, or an outcome that is between two values
Tutorial notes
- Sample distribution (here you take only one sample) VS sampling distribution (here you take
thousands of samples)
• Units of analysis: sample distribution a single candy VS sampling distribution a bag of candies
- Population = the overall units of analysis that we want to analyse (eg. Female nurses that work
in Amsterdam hospitals)
- Sampling space = numbers at the bottom of the sampling distribution
- If the population is bigger than the unbiased estimator = downward bias
- Probability density function = is the curve that forms when you sketch the probability density
(y-axis) on the graph, sampling space (x-axis)
- Right-hand (threshold) vs left-hand (threshold) probability
Summary
- Statistical inference is about estimation and null hypothesis testing
- We collect data on a random sample and we want to draw conclusions (make inferences) about
the population from which the sample was drawn
- The sample does not offer a perfect image of the population (drawing another random sample
would have different characteristics)
- To make an informed decision on the CI we must compare the characteristics of the sample we
have drawn to the ones of the sample we could have drawn
- Sampling distribution = characteristics of the samples that we could have drawn
1.1 Statistical inference: making the most of your data
- Statistics is a tool for scientific research
- Collecting data is expensive
- Inferential statistics offers techniques for making statements about a larger set of observations
from data collected for a smaller set of observations
- The large set of observation about which we want to make a statement is called the population
- The smaller set is called a sample
- Statistical inference is a generalisation from the data collected in a random sample to the
population from which the sample was drawn
1.2 A discrete random variable: how many yellow candies in my bag?
Sample statistic
- We are usually interested in a particular characteristic of the sample rather than in the exact
nature of each observation within the sample
- Sample statistic: a number describing a characteristic of the sample
- Each sample has one outcome score on the sample statistic
- All possible outcome scores constitute the sampling space
- Sample statistic is a random variable
Sampling distribution
- Some sample statistic outcomes occur more often than other outcomes
- Sampling distribution = distribution of outcome scores of very many samples
Probability and probability distribution
- What is the probability of drawing a sample with five yellow candies as a sample statistic
outcome? This probability is the proportion of all possible samples that we could have drawn that
happen to contain five yellow candies
- Sampling distribution tells us all possible samples that we could have drawn
,- If we change the frequencies in the sampling distribution into proportions, we obtain the
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: because only a limited number of outcomes are possible
- Sampling distribution as a probabiloty distribution conveys very important information
• Tells us which outcomes we can expects and the probability that a particular outcome may
occur
Expected value or expectation
- If the share of yellow candies in the population is 0.20, we expect one out of each five candies in
a bag to be yellow
- The expected value of the proportion of yellow candies in the sample is equal to the proportion of
yellow candies in the population
- Expected value = the average of the sampling distributions of a random variable
- The sampling distribution is an example of a probabiloty distribution, the excepted value is the
average of a probability distribution
- The expected value is also called the expectation of a probability distribution
Unbiased estimator
- Expected value of the proportion of the sample statistic equal the true proportion of the
population statistic
- The sample proportion is an unbiased estimator of the proportion in the population
- Unbiased estimator = when expected value (mean of the sampling distribution) is equal to the
population statistic
- Parameter = population statistic
- Downward biased = it is too low (population much larger than the sample)
Representative sample
- A sample is representative of a population if variables in the sample are distributed in the same
way as in the population
- A random sample is likely to differ from the population due to chance, so the actual sample that
we have drawn is usually not representative of the population
- BUT we should expect it to be representative, so we say that it is in principle representative of
the population
1.3 A continuous random variable: overweight and underweight
Continuous variable
- Weight is a continuous variable because we can always think of a new weight between two other
weights
- If we can always think of a new value between two values, the variable is continuous
,Continuous sample statistic
- We are interested in the average weight of all candies in our sample bag, so average candy weight
in our sample is our key sample statistic
- The sample mean is an unbiased estimator of the population mean, so the average weight of all
candies in the population is the average of the sampling distribution
- And this is the average weight that we expect in a sample drawn from this population
Probability density
- We can solve this problem by looking at a range of values instead of a single value
- Choose a threshold and determine the probability of values above or below this threshold
- Probability density function = a curve, so if we cannot determine the probabiloty of a single
value, we display probabilities between the horizontal axis and this curve
- Probability density function = can gives us the probability of values between two thresholds
• It can also five us the probability of values up to (and including) a threshold value, which is
known as a left-hand probability or the probability of values above (and including) a
threshold value, which is called a right-hand probability
1.4 Concluding Remarks
Sample characteristics as observations
- Sampling distributions are our cases (units of analysis) and sample characteristics are our
observations
- In a sampling distribution, we observe samples (cases) and measure a sample statistic as the
(random) variable
- Each sample adds one observation to the sampling distribution and its sample statistic is the value
added to the sampling distribution
Means at three levels
- The mean of the sampling distribution is the average of the average weight of candies in every
possible sample bag
- This mean of means had the same value as our first mean, namely the average weight of the
candies in the population because a sample mean is an unbiased estimator of the population
mean
- The population and the sample consist of the same type of observations
Take-home points
- Values of a sample static vary across random samples from the sample populations. But some
values are more probable than other values
- The sampling distributions of a sample statistic tells us the probability of drawing a sample with a
particular value of the sample statistic or a particular min/max value
, - If a sample statistic is an unbiased estimator of a parameter, the parameter value is the average of
the sampling distribution, which is called the expected value or expectations
- For discrete sample statistics, the sampling distribution tells us the probability of individual
sample outcomes. For continuous sample statistics, it tells us the probability density, which gives
us the probability of drawing a sample with an outcome that is at least or at the most a particular
value, or an outcome that is between two values
Tutorial notes
- Sample distribution (here you take only one sample) VS sampling distribution (here you take
thousands of samples)
• Units of analysis: sample distribution a single candy VS sampling distribution a bag of candies
- Population = the overall units of analysis that we want to analyse (eg. Female nurses that work
in Amsterdam hospitals)
- Sampling space = numbers at the bottom of the sampling distribution
- If the population is bigger than the unbiased estimator = downward bias
- Probability density function = is the curve that forms when you sketch the probability density
(y-axis) on the graph, sampling space (x-axis)
- Right-hand (threshold) vs left-hand (threshold) probability
