Data distributions, z scores and p-values Flashcards
1
Q
What are distributions of data and how are data distributions commonly visualised?
A
- distributions of data are the manner in which data for a particular variable is spread over its range
- data distributions are commonly visualised using a histogram
2
Q
What does the shape of the histogram tell us?
A
- are there more common scores in our data set? (mode)
- Is there a range in which data is more concentrated
- Chances of randomly picking someone/something from the sample in a particular range (probability)
3
Q
What shape is a normally distributed histogram and what does this show?
A
- Bell curve
- It has a peak in the middle and trails off either side
- Shows that there’s an average (e.g. height) and then fewer people with lower heights and higher heights
4
Q
When can we find it hard to see normality in a histogram?
A
If we don’t have much data/ small sample size
5
Q
What are examples of Non-normally distributed data?
A
- Positively Skewed data has a tail to the left (reaction time)
- Negatively Skewed data has a tail to the left
- Danger: mean is distorted by tails! (median is a good way to get around that)
6
Q
what are features of bimodal data?
A
- Two distinct populations
- Two peaks
- The mean will be in the middle and not representative of the common score in the data
- Usually bimodal data indicated something has gone wrong with your experiment and you may actually have two populations
7
Q
what is the normal distribution specified by? (equation)
A
- Mean and standard deviation
- N(u,o) commonly used to describe
- u = mean
- o = standard deviation
- N = normality
8
Q
give features of normality distributed histograms
A
- Mean is the line down the centre of the curve
- Standard deviation is related to the width of the curve
- All are bell shaped
- symmetric about the centre
- Tails never reach zero (very close though)
- The area under the curve is always equal to 1
- Very close to 0 by the time it gets to 3 standard deviations away from the mean (e.g. mean +/- 3 standard deviations from mean)
9
Q
Why do we need to test probability?
A
- A major goal for us in using statistics is to be able to use our data to test experimental hypotheses
- We can never be certain about the validity of our hypotheses so probabilities will be fundamentally important
- Underlies Null Hypothesis testing
10
Q
What is probability?
A
- We can think of a probability as a measure of how likely it is that an uncertain event will occur
- Probabilities can be expressed as a percentage or proportion
- 0% = impossible
- 100% = certain
11
Q
What is the equation to work out the probability of an event occurring P (event)
A
P(event) = no. of possible outcomes consistent with event/No. of possible outcomes
12
Q
What is conditional probability and give an example
A
- Probability of an event given that something else is known/assumed, i.e. when given/assuming some other additional information
- E.g. I close my eyes and role 1 die. An honest observer tells me I have rolled a number <4. Before I open my eyes what’s the probability that I have rolled an even number?
No. possible outcomes = 3
Only 1 is even
So P(event) = 1/3
P(even <4) where the line means given
When you assume something the assumption is the same
13
Q
Give an example of working out area under data distributions for uniform distribution (where outcomes are equally likely)
A
- You have a friend who always arrives to meet you somewhere in the range from 5 minutes before to 5 minutes after the agreed meet time. They are never earlier than 5 mins before and never later than 5 mins after and time they arrive is completely random within that range
- You keep track of arrival times for 100 meetings in a year
- Q. Based on your sample of data what is the probability that your friend is at least two minutes late?
- A. Friend was late <2 minutes 29 times out of 100 meetings so the probability will be 29/100
- Useful to express as a proportion:
- Divide each by 100 – total of all bars is 1
- If all the bars = 1 then if we look at the bars in our range of interest then the bar heights themselves as a proportion of 1 will give us our proportion
- Remember your friend is equally likely to arrive at any time in the range from 5 minutes before to 5 minutes after the agreed meeting time
- So the histogram should look uniform (it was just messed up by a smaller sample size
- Area of distribution tells us about the probability
14
Q
What is a z score (including equation)?
A
- the z score is obtained by subtracting the population mean from x and then dividing by the standard deviation
- X-u shows how far away your score is from the mean
- Diving by u shows what proportion this is of the standard deviation
- E.g. for someone’s IQ 120-100/15 -1.33. This shows us that our score is 1.33 standard deviations higher than the mean
15
Q
Give features of z-transformation in general
A
- will take any normally distributed data and convert to z score
- if you find out z for all scores you will get a normal distribution with mean 0 and standard deviation 1
- N(0,1)
- this means there is one standard normal distribution that all data will follow
- Therefore by matching data for this with values in a chart we can work out the area (which tells us the probability)
