Ch6 - The Normal Curve, Standardization, and z Scores Flashcards
What can the normal curve tell us?
- Allows us to determine probabilities about data and then draw conclusions that we can apply beyond the data
Relationship between the data set + size:
- As the data set increases, the distribution more and more closely resembles a normal curve (central limit theorem)
- AKA, as the size of the sample approaches the size of the population, the shape of the distribution tends to be normally distributed
Why can scientists use the normal curve to make meaningful comparisons?
When data are normally distributed, we can compare one particular score to an entire distribution of scores (such as a z score)
To compare one particular score to an entire distribution of scores, we convert…
- …one raw score into a standardized score
Standardization
- a way to convert individual scores from different normal distributions to a shared normal distribution with a known mean, standard deviation, and percentiles
- like changing different measurements to the same unit
What is one of the first problems with making meaningful comparisons?
variables are measured on different scales (EX: measuring height in inches but weight in kilograms)
How can we turn variables into the same scale?
- We can standardize different variables by using their means and SDs to convert any raw score into a z score
- z score: the number of standard deviations a particular score is from the mean
A z score is part of its own distribution, the…
- z distribution (just as a raw score is part of its own distributions)
- EX: a person’s height is part of its own distribution, a distribution of heights
What information do we need to convert any raw score -> z score?
- The mean
2. SD of the population of interest
2 important figures of the z distribution:
1: the z distribution always has a mean of 0
- If you’re exactly at the mean = you’re 0 standard deviations from the mean
2: the z distribution always has a SD of 1
- If your raw score is 1 SD above the mean, then you have a z score of 1
To calculate a particular z score
Z = X-μ / σ
Formula to calculate the raw score from a z score:
X = z(σ) + μ
OVERALL - what can we do with the mean and SD of a population?
- calculate the raw score from its z score
- calculate the z score from its raw score
The normal curve also allows us to convert scores into… + WHY?
percentiles, because 100% of the population is represented under the bell-shaped curve
* Thus, the midpoint is the 50th percentile
How can we make even more specific comparisons?
we convert raw scores to z scores and z scores to percentiles using the z distribution
The z distribution
- A normal distribution of standardized scores - a distribution of z scores
- Versus the standard normal distribution: a normal distribution of z scores
The standard z distribution allows us to do the following:
- Transform raw scores into standardized scores called z scores
- Transform z scores back into raw scores
- Compare z scores to each other - even when the underlying raw scores are measured on different scales
- Transform z scores into percentiles that are more easily understood
What % of scores fall between the mean and a z score of +/- 1
- 34% on either side of +1/-1
- 68% in total
What % of scores fall between the z scores of +/- 1 and 2?
- 14% between +/- 1 to 2
- IN SUM with the % between 0 and +/-1, 68% + 14% = 96% in total
What % of scores fall between the z scores of +/- 2 and 3?
- 2%, so 4% total + 96% = 100%
The central limit theorem
- refers to how a distribution of sample means is a more normal distribution than a distribution of scores, even when the population distribution is not normal
- As a sample size increases, a distribution of sample means more closely represents a normal curve
The CLT demonstrates two important principles:
- Repeated sampling approximates a normal curve, even when the original population is not normally distributed
- A distribution of means is less variable than a distribution of individual scores
Distribution of means:
- distribution composed of many means that are calculated from all possible samples of a given size, all taken from the same population
- EX: in class, when we took the average of 3 numbers from a sample of over 200, and plotted them on a histogram
- AKA: the examples that make up the distribution of means are not individual scores, but rather, MEANS of samples of individual scores
Characteristics of distributions of means
- more consistently produces a normal distribution
- more tightly clustered than a distribution of scores
- not as many means at the far tails of the distribution as in the distribution of scores
Why does the spread decrease when we create a distribution of means rather than a distribution of scores?
- When we plotted individual scores, each extreme score was plotted on the distribution (all accounted for)
- However when we plotted means, we averaged each extreme score with two other scores
What would happen if you increase the distribution of means
- The distribution would be even narrower (higher peak on graph) because there would be more scores to balance the occasional extreme score
- THE LARGER THE SAMPLE SIZE, THE SMALLER THE SPREAD OF THE DISTRIBUTION OF MEANS
Why do we need a different SD for the distribution of means?
Because the distribution of means is less variable than the distribution of scores
μ
Distribution of scores - symbol for mean
σ
Distribution of scores - symbol for spread
μM
Distribution of means - symbol for mean
σM
Distribution of means - symbol for spread
Distribution of scores - name for spread
Standard deviation
Distribution of means - name for spread
Standard error
There’s a simple calculation that lets us know exactly how much smaller the standard error, om, is than the SD, o:
σM = σ/√N
3 important characteristics of the distribution of means
- As sample size increases, the mean of a distribution of means remains the same
- The SD of a distribution of means (called the standard error) is smaller than the standard deviation of a distribution of scores. As sample size increases, the SD error becomes even smaller
- The shape of the distribution of means approximates the normal curve of the distribution of the population of individual scores has a normal shape or if the size of each sample that makes up the distribution is at least 30
When we calculate the z score, we simply use a distribution of means instead of a distribution of scores:
z = (M - μM) / σM
