What is a probability distribution?
- A graph, table or formula that specifies the probability associated with each possible value the random variable can assume.
What does a probability distribution represent and what values can it hold?
- This probability distribution represents the theoretical population distribution.
- A probability distribution can have values that are a countable number of values (Discrete).
- Alternatively, a probability distribution can have values corresponding to any of the points in a interval or intervals (Continuous).
What symbols make up a Normal Probability Distribution?
x = Continuous random variable μ= Population Mean σ= Population Standard Deviation π = 3.14159, e = 2.71828 ~N(μ,σ)
What symbols make up a Standard Normal Probability Distribution?
z = Standardized x
z = (x - μ)/σ
~N(0,1)
Study the equations for a Normal Probability Distribution and a Standard Normal Probability Distribution
https://docs.google.com/document/d/1r_ttbYs-4jXdkBbVGPH9vk1swjRRmRJUWdllcJXdaAI/edit?usp=sharing
Provide a summary of a Standard Normal Distribution.
- A symmetrical, continuous “bell” shaped curve
- Common distribution for scale data
- Parameters of a Standard Normal distribution :
- μ = 0, μ = 1, or ~N(0,1)
- z values range from -∞ to +∞
- The z value is z standard deviations from a mean of zero (z=2)
What can be used to find the value of z in a Standard Normal Distribution?
- Again we have a table that can be used to obtain the probability of getting a certain value (z) within a distribution of Z values.
- The Standard Normal table gives the cumulated probability between the mean (0) and the z value
Study the use of the Standard Normal Table
https://docs.google.com/document/d/1r_ttbYs-4jXdkBbVGPH9vk1swjRRmRJUWdllcJXdaAI/edit?usp=sharing
Carry out the following example.
- Using the Standard Normal Table, find the area of Z from the mean (area goes from 0 to 1.90)
Answer: - First look up the area between the mean and 1.90 Standard Deviations (i.e. z=1.90)
- This can be represented as P(0
How would you find a negative z value using the Standard Normal Table?
- So far we have dealt with positive z values.
- However, when we have a negative z value, we can still look up the positive z value since the area is the same either side of the curve (symmetrical).
Give an example of how to find the area of a negative z value.
- Negative Z scores P(-1.2z>1.2)
= 0.3849
How do we find a total probability when given two points on the graph from the mean?
- We can add the two probabilities together to obtain a total probability
- For example: P(-1.42
How would you find the area if the probability didn’t start at the mean(0)?
- We have to use subtraction to get the answer
- Remember: P(0
How would you find the area if the probability didn’t start at the mean(0) but it does stop at a certain z point?
- Again, we are going to use subtraction to calculate our final probability
- For example: if the area that we are trying to find starts at 0.42 and ends at 1.18 on the graph, then we first need to find the values of the areas between the mean (0) and both of these points.
- P(0.42
How would you find the value of T when just given the probability?
P(T
What is a Log Normal Distribution?
- A continuous probability distribution of a random variable whose logarithm is normally distributed.
What is the difference between a Normal Distribution and a Log Normal Distribution?
Normal Distribution:
- Range - infinity to + infinity
- Symmetrical around mean
- No skewness
- Mean = Median = Mode
- Std. Dev. has no relationship with the Mean
Log Normal Distribution:
- Range 0 to + infinity
- Unsymmetrical around mean
- Right hand skewed
- Mode < Median < Mean
- Std. Dev. has a linear relationship with the Mean
- The larger the mean, the larger the Std. Dev.
- Graph of Log Normal distribution in google doc
What is the Central Tendency?
- In statistics, a central tendency is a central or typical value for a probability distribution.
How is the central tendency measured?
- Mode, median and mean are all meaningful measures of central tendency
How do we measure a variables distribution if it is in a single population?
- A single population has an unknown distribution of a random variable
- If a single sample of this population is being studied then the use of descriptive statistics is needed.
Useful descriptive statistics: - Compare mean, median, mode
- Skewness, kurtosis
Useful descriptive graphics (if n>30): - Box and Whisker plots
- Histogram
- Q-Q plot
How is a box and whisker plot beneficial?
- Box and Whisker plots can be useful in summarising the distribution of the data
What does unsymmetrical distribution of data in a box plot tell you?
- Unsymmetrical distribution of data around the median (i.e. skewed - most of the data is near zero) and a high number of extreme outliers are indicators of data that is not Normally distributed.
What does symmetrical distribution of data in a box plot tell you?
- Symmetrical distribution of data around the median (e.g. equal whiskers) and a low number of outliers are indicators of data that is Normally distributed.
How are histograms useful in determining distribution?
If a histogram appears to be: - Appears skewed - Appears unsymmetrical - Non-Normal distribution or if it: - Appears not to be skewed - Appears symmetrical - Possibly a Normal distribution
