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Lecture 7

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Carleton University

Psychology

PSYC 2002

Steven Carroll

Winter

Description

Lecture 7: distributions?
Distribution of the diced
- Let’s do some bad stats!
- When you roll a die, 6 things can happen:
• 1, 2, 3, 4, 5, 6
- This is our population of scores.
- What is the mean? 3.5
- What is the standard deviation? 1.71
Bad stats
- What z-score would you associate with a score of 3?
• Z = (3 – 3.5) / 1.71 = -.29
- What z-score would you associate with a score of 6?
• Z = (6 – 3.5) / 1.71 = 1.46
Bad stats
- What, according to the z-table in the back of the book, is the probability of rolling a score
less than or equal to “3” if the z-score for a “3” is -.29
• .3859
- What, according to the z-table in the back of the book, is the probability of rolling a score
less than or equal to “6” if the z-score for a “3” is 1.46
• .9279
Bad stats
- So, according to the z-table, you are more likely to roll a “4, 5, or 6” than a “1, 2, or 3”
- And, according to the z-table, there is 7.21% chance that you’ll roll something other than
a “1, 2, 3, 4, 5, or 6”
Huh?!
- Ok, so what’s wrong with what we did?
- We’re our calculations wrong?
- So wtf happened there?
I’ve been saying it all along
- What assumption is being made with regards to the z-table? • That the z-scores are normally distributed!!!
Distribution of the diced
- This is NOT a normal distribution
- This is a platykurtic distribution
- Changing these to z-scores doesn’t suddenly make the distribution normal
Let’s try something...
- Instead of just rolling the die once, let’s get a sample of size n = 2
• We’ll roll the die two times, and take the average for the roll
• We’ll get all possible averages
Sample size n = 2
- How many things can happen?
X = 36
Sample size n = 2
- What is the probability of obtaining a sample where the average = “3”?
} (1 + 5) / 2 = 6 / 2 = 3
} (2 + 4) / 2 = 6 / 2 = 3
} (3 + 3) / 2 = 6 / 2 = 3
} (4 + 2) / 2 = 6 / 2 = 3
} (5 + 1) / 2 = 6 / 2 = 3
Sample size n = 2
- What is the probability of obtaining a sample where the average = “3”?
• p ( M ≤ 3 ) = 5 / 36 = .1389 Now look at this:
M f
1 1
1.5 2
2 3
2.5 4
3 5
3.5 6
4 5
4.5 4
5 3
5.5 2
6 1
- What is the mean of this distribution of sample means?
• Hint: What is it between?
- What do you notice about this value?
And check this out...
And check this out... Wow!
- So what is the difference between the distribution of sample means when n = 1, n = 2,
and the distribution of sample means when n = 3?
n = 1
The central limit theorem
- For any population with a mean µ and a standard deviation , the distribution of sample
means for samples of size n will have a mean of µ and a standard deviation of / n and
will approach a normal distribution as n approaches infinity - Actually, by the time n = 30 the distribution is pretty normal
What does that mean?
- It means that we tend to do our stats on the means of samples
- The location of a score in a sample or in a population can be represented with a z-score
• Z = ( X - µ ) /
- BUT: researchers don’t want to study single scores
• We aren’t interested in one person or one particular rat. We’re interested in groups of
people or groups of rats
Definitions
- The distribution of sample means: a collection of sample means for all possible random
samples of a particular size (n) that can be obtained from a population
- Asampling distribution: a distribution of statistics obtained by selecting all of the
possible samples of a specific size from a population
• The distribution of sample means is an example of a sampling distribution
How we roll in the brain sciences
- Samples provide estimates of what is going on in the population! That’s good!
- Samples are variable. No two samples are identical. That’s bad!

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