Chapter 4 - Key Words Flashcards
Sum of X
The sum of the scores in a sample
Measures of Central Tendency
Mean, median, and mode
They call these the measures of central tendency because they TEND to MEASURE what is happening in the CENTER of the scores.
How to calculate the mode
Just count how many times each score appears, whichever score (or scores) appear(s) the most is the mode.
Median
The middle score (your book calls it the 50thpercentile)
How to calculate the median
Line up all the scores
Then cross out the highest score
Then cross out the lowest score
Then repeat until you have the middle score left
*If you’re using a distribution with an even number of scores, you need to calculate the average of the two middle scores and that will be the median.
How to calculate the mean
Add all the scores together, then divide by the number of scores in the data.
Range
The difference between the highest and lowest score in the data.
Let’s say that the highest score on a midterm exam is 97, and the lowest score is 58.
Sometimes it is given as one number, in this case it would be 39.
Sometimes it is given by simply reporting the lowest and highest scores, in this case it would be 58-97. I like this way better, because it gives more information without being too complex.
Deviation
A score’s distance from the mean.
These can be positive or negative values, and if one of your scores is exactly the same as the mean then its deviation would be 0.
Positive example: the mean is 55, your score is 61, so your deviation is 6.
Negative example: the mean is 30, your score is 19, so your deviation is -11.
Sample Standard Deviation
Sample standard deviation is the square root of sample variance.
Our sample variance from the last slide was 11.5, and the square root of 11.5 is about 3.39. We usually calculate to two decimal places. So if you were to answer “3” on the test, I wouldn’t be able to give you full credit.
Sum of the Squared deviations
Once we calculate all the deviations, we can “square” each deviation and then add all the “squares” together.
To “square” means to multiply a number by itself; 3 “squared” is 9.
Here are some scores: 3, 7, 10, 12
The mean is 8, and the deviations are -5, -1, +2, +4
The squares for these deviations are 25, 1, 4, 16 (remember that squaring a negative number gives you a positive result).
The sum of the squared deviations is 46.
Sample Variance
Sample variance is the sum of the squared deviations divided by the number of scores in the sample.
On the last slide, the sum of the squared deviations was 46, and the number of scores in the sample was 4.
46 divided by 4 is 11.5, so our sample variance is 11.5
Why did you just teach that to me?
Standard deviation is a very important concept in descriptive statistics (it helps us describe our data).
Variance is a very important concept in inferential statistics (it gets used in all the future tests/formulas we’ll talk about).
These two things are measures of how spread out our data is. Higher standard deviations mean the data are all over the place, lower standard deviations mean all the data are pretty close together.
Football Season
Let’s use a Pac-12 football season as our dataset. There are 12 Pac-12 teams. We’re going to count the number of total wins each team had and make a list of scores.
Okay, here were the 12 teams’ win totals, from lowest to highest:
4,4,4,5,5,6,8,8,8,8,11,12
Let’s turn it into a frequency distribution, y’know, a graph.
We’ll use a histogram or frequency polygon since win totals are interval/ratio data.
What does a larger measure of variability communicate about: the size of differences among the scores in a distribution?
There are larger and/or more frequent differences among the scores.
What does a larger measure of variability communicate about: how consistently the participants behaved?
The behaviors are more inconsistent.
What does a larger measure of variability communicate about: how spread out the distribution is?
The distribution is wider, more spread out.
In any research, why is describing the variability important?
What is the difference between what a measure of central tendency tells us and what a measure of variability tells us?
(a) What is the mathematical definition of the variance? (b) Mathematically, how is a sample’s variance related to its standard deviation and vice versa?
Why is the mean a less accurate description of the distri bution if the variability is large than if it is small?
How do we determine the scores that mark the middle 68% of a sample?
Find the scores a +1SX and at –1SX from the mean.
How do we determine the scores that mark the middle 68% of a known population?
Find the scores at +1σX and at –1σX from μ
How do we estimate the scores that mark the middle 68% of an unknown population?
Use X¯ to estimate μ , and then find the scores at +1sX and at –1sX from μ
