Statistical Reasoning “He told me I was average. I told him he was mean.”

Descriptive Statistics Used to organize and summarize data in a meaningful way. Frequency distributions – Where are the majority of the scores? Used to organize raw scores, or data, so that information makes sense at a glance. They take scores and arrange them in order of magnitude and the number of times each score occurs.

Histograms & Frequency Polygons (showing you data a glance) Ways of showing your frequency distribution data. 1.Histogram – graphically represents a frequency distribution by making a bar chart using vertical bars that touch When you have a continuous scale (for example, scores on a test go from 0-100, continuously getting larger.) the bars touch, because you have to have a class for each score to fall into, and you can’t have any “gaps.” Different than a simple Bar Graph which is used when you have non-continuous classes (example, which candidate do you support, Obama or McCain? You’d have a bar for each, with gaps in between, because you can’t fall between two candidates, you have to pick one.)

Histogram Uses a Bar Graph to show data

Frequency Polygon Uses a line graph to show data 2. Frequency Polygon – graphically represents a frequency distribution by marking each score category along a graph’s horizontal axis, and connecting them with straight lines (line graph)

Measures of Central Tendency A single number that gives us information about the “center” of a frequency distribution. Measures of central tendency – 3 types 4, 4, 3, 4, 5 1.Mode=most common=4 (Reports what there is more of – Used in data with no connection. Can’t average men & women.) 2. Mean=arithmetic average=20/5=4 (has most statistical value but is susceptible to the effects of extreme scores ) 3. Median=middle score=4 (1/2 the scores are higher, half are lower. Used when there are extreme scores)

Central Tendency An extremely high or low price/score can skew the mean. Sometimes the median is better at showing you the central tendency TOPPS Baseball Cards Nolan Ryan$1500 Billy Williams$8 Luis Aparicio$5 Harmon Killebrew$5 Orlando Cepeda$3.50 Maury Wills$3.50 Jim Bunning$3 Tony Conigliaro$3 Tony Oliva$3 Lou Pinella$3 Mickey Lolich$2.50 Elston Howard$2.25 Jim Bouton$2 Rocky Colavito$2 Boog Powell$2 Luis Tiant$2 Tim McCarver$1.75 Tug McGraw$1.75 Joe Torre$1.5 Rusty Staub$1.25 Curt Flood$1 With Ryan: Median=$2.50 Mean=$74.14 Without Ryan: Median=$2.38 Mean=$2.85

Does the mean accurately portray the central tendency of incomes? NO! What measure of central tendency would more accurately show income distribution? Median – the majority of the incomes surround that number.

Measures of Variability Gives us a single number that presents us with information about how spread out scores are in a frequency distribution. (See example of why this is important). Range – Difference b/w a high & low score –Take the highest score and subtract the lowest score from it. (can be skewed by an extreme score) Standard Deviation – How spread out is your data? –The larger this number is, the more spread out scores are from the mean. –The smaller this number is, the more consistent the scores are to the mean

Calculating Standard Deviation How spread out (consistent) is your data? 1.Calculate the mean. 2. Take each score and subtract the mean from it. 3.Square the new scores to make them positive. 4.Mean (average) the new scores 5.Take the square root of the mean to get back to your original measurement. 6. The smaller the number the more closely packed the data. The larger the number the more spread out it is.

Standard Deviation Punt Distance Mean: 160/4 = 40 yds Deviation from Mean = – 40 = – 40 = – 40 = +5 Deviation Squared Numbers multiplied by itself & added together Variance: 46/4 = 11.5 Standard Deviation: variance= 11.5 = 3.4 yds

Multiple Choice Essay Composite Mean=34.3 SD=4.2 Mean=10.2 SD=2.0 Mean=9.3 SD=2.3 Are these scores consistent? Is there a skew?

Z-Scores A number expressed in Standard Deviation Units that shows an Individual score’s deviation from the mean. Basically, it shows how you did compared to everyone else. + Z-score means you are above the mean, – Z-score means you are below the mean. Z-Score = your score minus the average score divided by standard deviation. Which class did you perform better in compared to your classmates? Test TotalYour Score Average score S.D. Biology Psych Z score in Biology: = 8, 8 / 4 = +2 Z Score Z score in Psych: = 6, 6/2 = +3 Z Score You performed better in Psych compared to your classmates.

9/14/ Photo courtesy of Judy Davidson, DNP, RN

Standard Normal Distribution Curve Characteristics of the normal curve Bell shaped curve where the mean, median and mode are all the same and fall exactly in the middle + or - # Wechsler Intelligence Scores

Skewed Curves

Inferential Statistics Help us determine if our results are legit and can be generalized to the public Help to determine whether a study’s outcome is more than just chance events. Used to predict things about a population based on a sample. 3 Principles of Inferential Statistics: 1.Non-biased sample - Representative Samples are better than biased samples for generalizing data 2.Less-variability is better – the average is better when it comes from scores of low variability 3.More cases are better than fewer – averages based on many cases are more reliable.

Statistically Significant Possibility that the differences in results between the experimental and control groups could have occurred by chance is no more than 5 percent Must be at least 95% certain the differences between the groups is due to the independent variable

Statistical Significance p value = likelihood a result is caused by chance. In other words, are they statistically significant? If the answer is yes, then they can be generalized to a larger population Researchers want this number to be as small as possible to show that any change in their experiment was caused by an independent variable and not some outside force. Results are considered statistically significant if the probability of obtaining it by chance alone is less than.05 or a P-Score of 5%. p ≤.05 Researcher must be 95% certain their results are not caused by chance. Replication of the experiment will prove the p value to be true or not. Effect Size – Measure of the strength of a relationship between variables (used with SS to report quality of results)

p Value Describes the percent of the population/area under the curve (in the tail) that is beyond our statistic This means the percentage of chance that a confounding variable may be responsible for our results. Check out P Values made simple for more help.P Values made simple