Ch6 - The Normal Curve, Standardization, and z Scores Flashcards

1
Q

What can the normal curve tell us?

A
  • Allows us to determine probabilities about data and then draw conclusions that we can apply beyond the data
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2
Q

Relationship between the data set + size:

A
  • 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
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3
Q

Why can scientists use the normal curve to make meaningful comparisons?

A

When data are normally distributed, we can compare one particular score to an entire distribution of scores (such as a z score)

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4
Q

To compare one particular score to an entire distribution of scores, we convert…

A
  • …one raw score into a standardized score
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5
Q

Standardization

A
  • 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
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6
Q

What is one of the first problems with making meaningful comparisons?

A

variables are measured on different scales (EX: measuring height in inches but weight in kilograms)

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7
Q

How can we turn variables into the same scale?

A
  • 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
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8
Q

A z score is part of its own distribution, the…

A
  • 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
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9
Q

What information do we need to convert any raw score -> z score?

A
  1. The mean
    2. SD of the population of interest
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10
Q

2 important figures/rules of the z distribution:

A

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

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11
Q

To calculate a particular z score

A

Z = X-μ / σ

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12
Q

Formula to calculate the raw score from a z score:

A

X = z(σ) + μ

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13
Q

OVERALL - what can we do with the mean and SD of a population?

A
  1. calculate the raw score from its z score
  2. calculate the z score from its raw score
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14
Q

The normal curve also allows us to convert scores into… + WHY?

A

percentiles, because 100% of the population is represented under the bell-shaped curve
* Thus, the midpoint is the 50th percentile

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15
Q

How can we make even more specific comparisons?

A

we convert raw scores to z scores and z scores to percentiles using the z distribution

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16
Q

The z distribution

A
  • A normal distribution of standardized scores - a distribution of z scores
  • Versus the standard normal distribution: a normal distribution of z scores
17
Q

The standard z distribution allows us to do the following:

A
  • 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
18
Q

What % of scores fall between the mean and a z score of +/- 1

A
  • 34% on either side of +1/-1
  • 68% in total
19
Q

What % of scores fall between the z scores of +/- 1 and 2?

A
  • 14% between +/- 1 to 2
  • IN SUM with the % between 0 and +/-1, 68% + 14% = 96% in total
20
Q

What % of scores fall between the z scores of +/- 2 and 3?

A
  • 2%, so 4% total + 96% = 100%
21
Q

The central limit theorem

A
  • 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
22
Q

The CLT demonstrates two important principles:

A
  1. Repeated sampling approximates a normal curve, even when the original population is not normally distributed
  2. A distribution of means is less variable than a distribution of individual scores
23
Q

Distribution of means:

A
  • 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
24
Q

Characteristics of distributions of means

A
  • 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
25
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
26
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
27
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
28
μ
Distribution of scores - symbol for **mean**
29
σ
Distribution of scores - symbol for spread
30
μM
Distribution of means - symbol for mean
31
σM
Distribution of means - symbol for spread
32
Distribution of scores - name for spread
Standard deviation
33
Distribution of means - name for spread
Standard error
34
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
35
3 important characteristics of the distribution of means
1. **As sample size increases, the mean of a distribution of means remains the same** 2. **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 3. **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**
36
When we calculate the z score, we simply use a distribution of means instead of a distribution of scores:
z = (M - μM) / σM