1. STATISTICS

2. SCALES OF MEASUREMENT
There are four basic scales of measurement. All scales of measurement fit into one of these four basic types.
Nominal Scale: uses numbers (or other symbols) to simply name objects or events that exist in the study (i.e. a table listing all subjects in a study)
Ordinal Scale: uses numbers (or other symbols) to not only name objects or events, but also to distinguish the events in terms of direction (lesser than or greater than) without determining how much of a difference exists between events(i.e. T-shirt sizes).
Interval Scale: has all the properties of the previous two scales, but also allow evaluations of how much difference exists between objects or events in the scale (i.e. Celsius scale). Offers an arbitrary zero point.
Ratio Scales: have all the properties of the previous three scales, but the difference is that a ratio scale has an absolute zero point, meaning there are no events less than 0 (i.e. measuring how tall you are by meters/feet).

3. Statistics
Statistics – using mathematics to organize, summarize/describe, and interpret numerical data so as to make inferences and predictions about phenomena.
Statistics are a part of everyday modern life…batting averages, economic projections, popularity ratings for TV shows, etc.
There are two basic types of statistics, descriptive and inferential.
Descriptive statistics are used to organize and summarize data to provide some sort of overview.
Inferential statistics use the laws of probability to allow researchers to interpret data and draw conclusions.

4. SAMPLING
When studying a population, a researcher usually cannot study everyone or everything in that population. In most cases, there’s simply not enough manpower, money, or time to do so. As a result, they will attain a much smaller cross- section of the population to study, and then make the claim that, since the sample behaved this way, they can extend their findings to the entire population. This subset of the population that is actually studied is called a sample. Once they conclude their study on the sample, the researchers will then attempt to generalize their findings from the sample to the population they are studying. The key concern of the researcher is to ensure that the sample group is as similar as possible to the overall population they are trying to generalize to. If it is, the sample is said to be “representative” . There are two sample gathering methods you need to know about.

5. SAMPLING
Random Sampling occurs when every subject in the population has an equal chance of being selected for the study. The flaws of this system should be obvious immediately. Some randomly selected people may not participate for one reason or another, others may not be readily identified. Even if you can avoid those problems, does a small group truly represent the entire population? The answer is obviously no. If you use a sample, you automatically introduce bias into your study. Unfortunately, the only truly representative sample is one where you study every member of the population. The best way to increase representativeness in a random sample is to increase the size of the sample.
Two criteria for a random sample:
Every member of the population must be identified
The selection process must truly be random
Problems with this method have to do with large populations and small sample sizes, which usually end with a very non-representative sample.

6. SAMPLING
Stratified Sampling: researcher predetermines certain demographic categories which are referred to as “stratifications”. Stratifications are categories of the sample the researcher wants to guarantee will be proportional to the overall population (i.e. dividing population into socioeconomic class stratifications, racial or ethnic stratifications, etc.). This is achieved by random sampling within each of the pre-identified stratifications.
In most cases, samples will be more representative with the stratified method, but no sample will ever be perfectly representative. As stated before, the only perfectly representative sample occurs only when the entire population is studied. Also, a researcher cannot possibly control for all stratifications, and it can be time consuming and costly when dealing with large target populations.

7. DESCRIPTIVE STATISTICS
Concerned with organizing and summarizing data. The primary techniques used to organize and summarize data are:
Frequency Distributions tell an observer how many times a certain event occurred. The following are types of frequency distributions:
Frequency Table/Chart
Frequency Polygon
Histogram

11. DESCRIPTIVE STATISTICS
Measurements of Central Tendency
Mean: The arithmetic average of a distribution of scores.
Median: The midpoint of a ranked distribution of scores.
Mode: The most frequently occurring score in a distribution of scores.
Measurements of Variability
Range: The difference between the highest and lowest score in a distribution of scores.
Variance : The average squared deviation from the mean in a distribution of scores.
Standard Deviation : The average deviation from the mean in a distribution of scores.

12. Descriptive Statistics: Variability
Variability refers to how much scores vary from each other and from the mean. The standard deviation is a statistic that demonstrates the average deviation from the mean of a distribution. If there is high variability in a data set, there will be a high standard deviation. If there is low variability in a data set there will be a low standard deviation.
Which of the following distributions has a higher standard deviation?
0, 1, 3, 6, 7, 7, 8, 10, 11, 12
3, 8, 14, 19, 27, 34, 45, 58, 64, 68
It would be the second one, as there is a much greater amount of variation in the scores. In the first distribution, the scores are very close to one another, thus there is much lower variability.

13. DESCRIPTIVE STATISTICS
Using the concepts we just went through, we can now take data about a distribution and transform it onto a different scale so we can understand it from a different angle.
Z Scores: transforming a raw score on the original scale of measurement into a standard deviation score (tells an observer exactly how far from the mean a single score is in standard deviation units).
The Normal Curve: a theoretical scale which places scores into a perfectly symmetrical distribution, where an equal number of scores are on each side (above and below) the mean. It’s primarily used to transform a non-normal distribution into a normal distribution for the purposes of categorizing and comparing a set of scores.

14. Figure 2.11 Measures of central tendency

16. Correlation: Prediction, Not Causation
Higher correlation coefficients increase a researcher’s ability to predict one variable based on the other. For example, SAT/ACT scores are moderately correlated with first year college GPA . However, high GPA at college graduation is strongly correlated with initial starting salary. Therefore, the SAT is an ok predictor of college success, but GPA is an excellent predictor of starting salary. Remember always that correlation does not prove causation!!! Although correlation may allow prediction and even imply a causal relationship, it does not prove cause-and-effect. For example, a strong positive correlation has been shown between foot size in children and vocabulary…as foot size increases, so does vocabulary. Do bigger feet make children learn more words? No, it’s a third variable, age, which causes both feet and vocabulary to grow.

17. Correlation Coefficients
Correlational research allows us to demonstrate if there is a relationship of some sort between two variables. If we want to try and determine the strength of that relationship and what type of relationship it is, we must use a mathematical process that gives us a statistic called a correlation coefficient. Coefficients have several properties of importance. First of all, the number is always between 1 and A strong positive correlation will yield a coefficient that is close to 1, while a strong negative correlation coefficient will be a number close to -1. If the coefficient is close to 0, then there is little if any relationship evident between the variables.
Positive correlations occur when the two variables rise or fall in the same direction; negative correlations occur when the variables move in opposite directions.

18. Figure 2.13 Positive and negative correlation

19. Inferential Statistics: Interpreting Data and Drawing Conclusions
Researchers use inferential statistics to determine whether their data support their hypotheses. With these statistical methods, they can interpret data and draw conclusions.
Inferential statistics use the laws of probability to allow researchers to determine how likely it is that their findings are real, that is, not due to luck or chance.
Statistical significance is said to exist when the probability that the observed findings are due to chance is very low. Many psychologists define “very low” as fewer than 5 chances in 100 that results are not real. This is a .05 level of significance.

20. DESCRIPTIVE STATISTICS
Scatterplot/Scatter Diagram: A graphical representation of a distribution’s scores on both of a correlational study’s variables. One variable will make up the vertical axis of the graph, while the other will be graphed on the horizontal axis.

21. DESCRIPTIVE STATISTICS:TRANSFORMING DATA

22. DESCRIPTIVE STATISTICS:TRANSFORMING DATA

23. DESCRIPTIVE STATISTICS:TRANSFORMING DATA

24. DESCRIPTIVE STATISTICS:TRANSFORMING DATA

25. DESCRIPTIVE STATISTICS:TRANSFORMING DATA

26. DESCRIPTIVE STATISTICS
Skewed Distribution: a set of scores that are not symmetrical; in other words, a set of scores has more high than low scores or more low than high scores.
Positive skew is indicated when there are many more low scores than high scores in the distribution (SAT scores of incoming freshman at Dade-North).
Negative skew is indicated when there are many more high scores than low scores in the distribution (SAT scores of incoming freshman at Harvard).

27. DESCRIPTIVE STATISTICS
Percentiles: a statistic that provides information about the relative position of a single score compared to the other scores in the distribution. A percentile indicates the % of scores in the distribution that are at or below an individual score. If a get a 92 on an test, and the teacher says I did well, as I was in the 97th percentile. What that tells me is that 97% of all other scores on that test are either equal to or below my score. Can be plotted visually on a normal curve.