Statistical Reasoning in Everyday Life
The Need for Statistics
Off-the-top-of-the-head statistics usually mislead the public. However, accurate statistical understanding benefits everyone. To be an educated person today is to simply be able to memorize statistics and apply them to everyday reasoning.
Descriptive Statistics
Descriptive Statistics: numerical data used to measure and describe characteristics of groups. Includes measures of central tendency and measures of variation.
Histogram: a bar graph depicting a frequency distribution.
This is used to measure and describe characteristics of the group under the study. Teachers use descriptive statistics to assess how their student have performed.
Measures of Central Tendency
The next step is to summarize the data using some measure of central tendency. A single score that represents a whole set of scores.
Mode: The most frequently occurring scores in a distribution
Mean: The arithmetic average of a distribution, obtained by adding the scores and then dividing by the number of scores.
Median: The middle score in a distribution, half below and half above.
Skewed Distribution: A representation of scores that lack symmetry around their average value.
Measures of central tendency neatly summarize data, but consider what happens to the mean when a distribution is lopsided, when itās shewed by a few way-out scores. The mean, mode, and median all tell completely different stories because the mean is biased by a few extreme outcomes.
Measures of Variation
Averages derived from scores with low variability are more reliable than averages based on scores with high variability. Consider a basketball player who scored between 13-17 points in each of the seasonās first 10 games. Knowing this, we would be more confident that she would score near 15 points in her next game than if her scores had varied from 5 to 25 points.
Range: the difference between the highest and lowest scores in distribution.
Standard Deviation: A computed measure of how much scores vary around the mean score.
Normal Curve: A symmetrical, bell-shaped curve that describes the distribution of many types of data; most scores fall near the mean and fewer near the extremes.
Inferential Statistics
Inferential Statistics: Numerical data that allow one to generalize- to infer from sample data the probability of something being true of a population.
How confidently, can we infer that an observed difference is not just a fluke-- a chance result from the research sample.
When Is an Observed Difference Reliable?
Research never randomly samples the whole human population. Keeping in mind what population a study has sampled is crucial.
An average is more reliable when it comes from scores with low variability.
Variability: Liability to vary or change. With the lowest and highest points not being so far away.
What Not to Do
An eager high school senior visits two university campuses, each for one day. On day one he views two classes from school number one and on day two he does the same but for the second school. He finds that in school one both teachers are emphatic to be teaching, but school two has teachers that donāt seem to be as excited. His review of the schools are not with a big enough sample size.
When Is an Observed Difference Significant?
When averages from two samples are each reliable measures of their respective populations, then their difference is probably reliable as well. When the difference between the sample averages is large, we have even more confidence that the difference between them reflects a real difference in their populations.
In short, when sample averages are reliable, and when the difference between them is relatively large, we say the difference has statistical significance.
Statistical Significance: A statistical statement of how likely it is that an obtained result occurred by chance.