distributions

Question 1

the binomial distributions

Answer 1

• one of the simplest distributions
• an idealised representation of the process that generates sequences of any process that gives rise to binary data
• its an idealisation but natural processes do give rise to binomial distribution

Question 2

normal distribution

Answer 2

• The binomial distribution has a shape that is similar to the normal distribution.
• But there are a few key differences:
  
  1. The binomial distribution is bounded at 0 and n(number of coins) - the normal distribution can range from + infinity to - infinity
  2. The binomial distribution is discrete (0,1,2,3 etc, but no 2.5) - normal distribution is continuous.
• The normal distribution is a mathematical abstraction, but we can use it as model of real-life populations that are produced by certain kinds on natural processes.

Question 3

processes that produce normal distributions

Answer 3

• to see how a natural process can give rise to a normal distribution
• There’s only 1 rule - you roll the dice n times (number of rounds), add up all the values, and move than many spaces. That is your score.
  
  • We can play any number of rounds
  • And we’ll play with friends, because you can’t get a distribution of scores if you play by yourself.
  • If we have enough players who play enough rounds then the distribution of scores across all the players will take on a characteristic shape.
  • A players score on the dice game is determined by adding up the values of each roll.
  • So after each roll their score can increase by some amount.
• The dice game might look artificial, but it maybe isn’t that different to some natural processes.
• For example, developmental processes might look pretty similar to the dice game.
• Think about height:
  
  • At each point in time some value can be added (growth) or a person’s current height.
  • So if we looked at the distribution of heights in the population then we might find something that looks similar to a normal distribution.
• A key factor that results in the normal distribution shape is this adding up of values.

Question 4

processes that dont produce normal distributions

Answer 4

• Let’s change the rules of the game:
• Instead of adding up the value of each roll, we’ll multiply them (e.g., roll a 1,2, and 4 and your score is 8).
• The distribution is skewed with most player having low scores and a few players have every high scores.
• Can you think of a process that operates like this in the real world?
  
  • How about interest or returns on investments?
• Maybe this explains the shape of real world wealth distributions.

Question 5

describing normal distributions

Answer 5

• the normal distribution has a characteristic bell shape but not all normal distributions are identical
• they can vary in terms of where they are centred and how spread out they are
• changing mean and standard deviation changes the absolute position of points on the plot, but not the relative positions measured in units of standard deviation

Question 6

describing deviations from the normal distribution

Answer 6

• When looked at the distribution of scores from the second dice game we saw that it was skew.
  
  • Skew is a technical term to describe one way in which distributions can deviate from normal.
  • This distribution has a skewness of 0. It is symmetrical.
  • Another way to deviate from the normal distribution is to have either fatter or skinnier tails.
  • The tailedness of a distribution is given by its kurtosis.
  • Kurtosis of a distribution is often specified with reference to the normal distribution. This is excess kurtosis.
    This distribution has an excess kurtosis of 0. It is a Mesokurtic distribution.

Question 7

distributions and samples

Answer 7

• we’ve seen that whenever we look at the distribution of values where the values are produced by adding up numbers we got something that looked like a normal distribution
• to calculate a sample mean, we just add up a bunch of numbers
• lets say i take lots of samples from a population, and for each sample, i calculate the sample mean

Question 8

the sampling distribution of the mean

Answer 8

• Population mean of 100
• And a standard deviation of 1.5
• From this population I can draw samples of 25 values.
• I’ll do this 100,000 times and plot the results in Figure 5.
• The standard deviation of the sampling distribution of the mean has a special name - the standard error of the mean.

Question 9

the central limit theorem

Answer 9

• You might think that the sampling distribution of the mean is normally distributed because the population is normally distributed
• But this is not the case, as your sample size increases, then sampling distribution of the mean will be normally distributed.
• And this will happen even if the population is not normally distributed.
• If the sample size is large enough, then the sampling distribution of the mean will approach a normal distribution. This occurs even if the population isn’t normally distributed.

Question 10

the standard error of the mean

Answer 10

• In lecture 7, we started talking about the spread of sample means around the population.
• I showed you figure 8, where the average deviation of sample means from the population mean was either small (A) or large (B).
  
  1. If the average (squared) deviation in the population is 0 then the average deviation of sample means from the population mean would be 0.
• Because all members of the pop would be the same, so all samples would be the same, so all sample means would be the same.
• The avergae (squared) deviations in the pop was larger, then the average deviations of sample means from the pop mean would be larger.
  
  2. If the sample size was large then the average deviation of sample means from the pop mean would be 0.
• Because every sample would be identical to the pop, so every sample mean would be identical to the pop mean.
• If the sample size was smaller, then the average deviations of sample means from the pop mean would be larger.
  Lets put these 2 ideas together to try come up with a formula for the average deviations of the sample means from the pop mean.
• There’s one final step to get to the formula for the standard error of the mean.
• The formula in Equation 3 is framed in terms of the average (squared deviations) of sample means from the pop mean - that is, in terms of variance.
• But the standard error of the mean is the standard deviation of the sampling distribution.
• The standard deviation is just the square root of the variance, so we just need to take the square root of both sides of equation 3, to get equation 4:
• The formula for the standard error of the mean and where it comes from.
  
  • This was, admittedly, a fairly long winded way to get to what is essentially a very simple formula
  • However, as I have alluded to several times, the standard error of the mean is a fairly misunderstood concept
  • I hope that getting there the long way has helped you to build a better intuition of what the standard error of the mean actually is
• I dislike talking about misconceptions because I think it can sometimes create them
• But it worth talking about one prominent one
  
  • Misconception
• The SEM tells you how far away the sample mean is (likely) to be from the actual population mean
• But it doesn’t tell you anything about the sample mean… at least not your sample mean that you have calculated for your particular sample
• The standard error of the mean is just what we’re defined it as:
  
  • The standard deviation of the sampling distribution
• So what does this tell you?
  
  • It tells you how far on averages sample means (not your sample mean) will be from the population mean
  • Your sample mean might be close to the population mean, it might be far away from the population mean. But the SEM doesn’t quantity this
• Your sample mean is either close or it is far from population mean
  
  • The SEM tells you something about the consequences of a sampling process
• Not something about your sample