Ch 6 - The Normal Curve Flashcards
Explain how the word ‘normal’ is used by statisticians.
- The normal curve is a bell-shaped & symmetrical curve.
- It also describes the distribution of many variables (for Psychology, many psychological variables!)
- Can use the area under the curve to determine the proportion of scores
in the normal distribution, the mean & mode & median are all in the direct middle.
What point on the normal curve represents the most commonly occurring observation?
that as the sample size increases, the shape of the distribution becomes more like the normal curve
The function depends on the mean & standard deviation
SMALL standard deviation : tall & skinny
LARGE standard deviation : short & wide
How does the size of a sample of scores affect the shape of the distribution of data?
Law of Large Numbers, remember
If the underlying distribution is normal, the more samples we get, the more likely the distribution will approximate the normal curve
What is meant by ‘standardized’?
What is a distribution of means OR sampling distribution?
SAME THING
A probability distribution based on a large # of samples of size from a given population
Can use the Central Limit Theorem to tell us the mean and standard deviation of this distribution of means.
These are ‘Distribution of Scores’
estimate the probability of drawing any one number from the pack of cards, like in the 2nd graph, it will tell me I have a 24% of drawing a #7 card.
Distribution of Means
a distribution graph can be made for any statistic, but MOST OFTEN, it’s created for ‘means’ (see below)
Distribution of the Mean of the Means
This is the data that was used to make the Distribution of Means on the right side:
Why do the Means only go between 3 —> 6.5? Because you are more likely to have more chance of pulling a card near the MEAN…because that’s what the mean is. It’s the average card you’ll most likely pull out.
From Ch 2, any Frequency Table (see next card) or any ‘Grouped Frequency Table’ (graph below) can also be made into a Distibution of Means
Key Note: notice how the mean of the 20 samples (4.75) is still close to the 5 samples (5.05) from the previous cards. But the standard deviation of the original population was 1.73, opposed to the s.deviation from below, of 0.82.
A Distribution of Means will always have a lower standard deviation than the original population BECAUSE there is less variability.
The values are being pulled towards the mean, so the means will have less variability in the distribution table.
Key Note : could also use like a Distribution of Scores to predict how often I’d get a mean of between 5 - 5.49….answer is 20%
An example of a Frequency Table being made into a Distribution Graph
What is a z-score?
Can turn any normal curve into a standard normal curve (aka. Z-CURVE)
What does a z-statistic (a z-score based on a distribution of means) tell us about a sample mean?
Give 3 reasons why z scores are useful
Point 1. Many random variables produce normal distributions (IQ, height, brain weights, stock market risks). We can analyse raw scores. (see card #7 for example)
Point 2. Standardization (allows us to compare scores) (see card #9 for example)
Point 3. Inferential Statistics (by finding a score on a normal curve, we can determine the likelihood of obtaining that score)
- ALSO, they will also show if they are (-) or (+) AND how far away from the mean they are.
EXAMPLE #1: computing a z-score
EXAMPLE #2: reverse – getting a raw score from a z-score
EXAMPLE #3: comparing variables-
Calculating ‘Percentile’
Why is the central limit theorem such an important idea for dealing with a population that is not normally distributed?
A distribution of means will be more likely to have a lower variance than a distribution of scores.
Why?
Remember the Central Limit Theorem states that the mean will be the same as the distribution of scores (not higher or lower). It also states that the standard error (standard deviation of the distribution of means) will be the standard deviation of the distribution of scores divided by the square root of N. That means the variance and the standard deviation will ALWAYS be smaller than the variance (or the standard deviation of a distribution of scores).
