The Basics
Scales of Measurement
There are four ways to measure data. The method used helps statisticians determine which statistical analysis to use.
Nominal measurement is the lowest level of measurement—the naming level. Some examples of nominal data are political parties, gender, and state. This data is not rankable—for example, you can’t say that Maryland (as a name) ranks higher or lower than Virginia.
Ordinal measurement puts participants in order from high to low, but this rank doesn’t indicate now much higher or lower one subject is from another. An example of this could be if a researcher wanted to rank participants by height: the shortest subject receives a rank of 1, the second shortest gets a rank of 2, etc. Note that the difference in height between the participants ranked 1 and 2 is not necessarily the same as the difference betwen those ranked 2 and 3. (Their heights might be, respectively, 5 feet, 5 feet 1 inches, and 6 feet.
Interval measurement uses a set scale, so the researcher knows both the order of the data points in relation to each other and on absolute terms, but it has no set zero. An example of this is temperature: we know that 32 degrees is ten degrees colder than 42 degrees, but a temperature of zero degrees doesn’t mean that there is no temperature. Interval data has a meaningful average.
Ratio measurement is a lot like interval, and both use the same statistical tests. However, ratio data does have an absolute zero. An example of this is test scores: zero has a meaning; it means there are no correct answers on the page. Ratio data has a meaningful average.
The level of measurement is not inherent to the variable itself. You can take the same concept/variable and measure it in a way that meets each of these definitions. For example, height could be nominal if you grouped heights into randomly assigned numbers (for example, group A is composed of people that are 5'0"-5'5" tall), or ordinal, or ratio.
Helpful Terms:
N: The number of participants
Frequency (f): How often a score occurs in a set of data
Range: The lowest and highest points of a set of data.
Cumulative Frequency: How many numbers are in and below a given score interval
Histogram: Essentially a graph of the frequency of a data set
Frequency polygon: Finding the midpoints in a frequency distribution and then connecting the points with a line. The shaped formed is the polygon. The more cases, and the smaller the bins, the more curved the polygon will look.
Distributions:
Normal: In a perfect (normal) distribution, the mean, median, and mode all have the same value 
Skewed:
In a negative skew, the mode is at the top of the curve, the median is lower than it, and the mean is lower than the median. 
In a positive skew, the mode is at the top of the curve, the median is higher than it, and the mean is higher than the median. An easy way to remember “positive” and “negative” skews is to look at the tail of the graph—if the tail is longer towards zero (or more negative), it is a negative skew. If the tail is longer towards the maximum (or more positive), it is a positive skew. 
Bimodal
When your data has two modes, creating a dip in the middle. This graph resembles a camel’s back.
Measures of central tendency:
Measures of central tendency help to determine how much the numbers in a distribution vary from each other and how much they tend towards some hypothetical middle.
Mean: In a range of scores, the mean is the sum of all scores divided by the number of participants. It requires that the data be either interval or ratio. It can be symbolized as an M or an X with a bar over it
Median: In a range of scores, the median is the number that falls exactly in the middle, such that 50% of the cases fall above it, and 50% fall below it. The data must be ordinal, interval, or ratio. The median is used to determine the inter-quartile range.
Mode: This term refers to the number that occurs most often in a data set. It can be used with any kind of data. A distribution can have more than one mode.
Measures of Variability
Deviation: How far a score is from the mean
Inter-quartile range: Takes the middle 50% of the scores, the median lying in the center. This is a good estimator of variability if there are outliers that skew the mean
Standard deviation: On average, how far the scores are from the mean. In a normal distribution, 68% of the cases will fall 1 standard deviation above and below the mean. 95% will fall 2 standard deviations above and below the mean. 99.7% will fall three standard deviations above and below the mean.
Note: see the excel tutorial to learn how to compute measures of central tendency on excel.
Probability, Normal Curve
Review Distributions
Probability is the number of times something is likely to occur out of the total number of possibilities.
Principles of Probability
1. If something is certain to occur, the probability is 1.00
a. Example: the probability that a number will be rolled on a dice is 1.00
2. If something is certain to not occur, the probability is 0.00
a. Example: the probability of rolling a letter on a numbered dice is 0.00
3. Mutually exclusive events are events that are the result of chance. They are independent of each other and cannot both happen at once.
a. Example: Rolling a six on the dice, then rolling a two. The chances of rolling either are 1 out of 6. On the second roll, the chances of rolling a two (1 out of 6) do not change. You cannot roll a six and a two simultaneously with only one dice.
4. Additive rule: The sum the probabilities of separate (mutually exclusive) events to determine the probability that any of them will occur
a. Example: The odds that a six or a two will be rolled is (1/6) + (1/6) or 2/6.
5. Multiplicative rule of independent events: The probability of the joint occurrence of two or more independent events is the product of their individual probabilities
a. Example: the probability of rolling a six and then a two is (1/6) * (1/6) or 1/36
6. Conditional probability: the probability that one event will occur given that some other event has occurred.
a. Example: if the number of possibilities changes with each successive drawing, e.g. picking numbers from a bag:
i. There are six tiles in a bag numbered one through six. The probability of picking a six is 1/6. If this tile is picked and removed, there are five tiles left. The probability of picking a two is now 1/5, not 1/6.
ii. To calculate the odds of drawing both: the probability of picking a six, removing it, and then picking a two is (1/6) * (1/5) or 1/30
Remember, regardless of the shape of the population from which the sample was drawn, the sampling distribution of all means tends to be normal. Therefore, the probability is not difficult to determine: 99.7% of all data points fall within three standard deviations of the mean for a standard normal distribution. This principle can be used to determine the probability of something happening.
Back to Statistics
References:
Patten, Mildred L. (2002). Understanding research methods: An Overview of the essentials (3rd ed.). Los Angeles: Pyrczak Publishing.
Pyrczak, Fred. (2002). Success at statistics: A Worktext with humor (2nd ed.). Los Angeles: Pyrczak Publishing.
Schutt, Russell K. (1999). Investigating the social world: the Process and practice of research (2nd ed.). Thousand Oaks: Pine Forge Press.
Solso, Robert L., Johnson, Homer H., & Beal, M. Kimberly. (1998). Experimental psychology: a case approach (6th ed.). New York: Longman.
