Descriptive Statistics

the everyday notions of central tendency Usual Customary Most Standard Expected normal Ordinary Medium commonplace NY Times, 10/24/ 2010 Stories vs. Statistics By JOHN ALLEN PAULOS

Overview What are descriptive statistics? A bit of terminology/notation Measures of Central Tendency Mean, Mode, Median Measures of Variability Ranges, Standard Deviations The Normal Curve

Terminology/Notation A data distribution = A set of data/scores (the whole thing) 1, 2, 4, 7 X = A raw, single score (i.e., 2 from above) ∑ = Summation (added up) ∑X = 14 (each individual score added up) n = sample size (distribution size, or number of scores) n = 4 (from above)

Descriptive Statistics Descriptive statistics are the side of statistics we most often use in our everyday lives Realize that most observations/data are too “large” for a human to take in and comprehend – we must “reduce” them How can we summarize what we see? Example – Grades/Registrar

Descriptive Statistics Descriptive statistics = describing the data n = 50, a test score of 83% Where does it fit in the class?? Making sense out of chaos

Descriptive Statistics Transform a set of numbers or observations into indices that describe or characterize the data “Summary statistics” A large group of statistics that are used in all research manuscripts Even the most complex statistical tests and studies start with descriptive statistics

Descriptive Statistics Measurement Scales Relationship Scatterplot Correlation Regression Nominal Ordinal Interval Ratio Descriptive Statistics Graphic Portrayals Variability Central Tendency Range Standard deviation Standardized scores Frequencies Histograms Bar graphs Normal distribution Mean Median Mode

Descriptive Statistics Descriptive statistics usually accomplish two major goals: 1) Describe the central location of the data 2) Describe how the data are dispersed about that point In other words, they provide: 1) Measures of Central Tendency 2) Measures of Variability

Measure of Central Tendency What SINGLE summary value best describes the CENTRAL location of an entire distribution? Mode: which value occurs most often Median: the value above and below which 50% of the cases fall (the middle; 50th percentile) Mean: mathematical balance point; arithmetic/mathematical average

Mode Most frequent occurrence What if data were? 17, 19, 20, 20, 22, 23, 25, 28 17, 19, 20, 20, 22, 23, 23, 28 Problem: set of numbers can be bimodal, or trimodal, depending on the scores Not a stable measure Ex. 17, 19, 20, 22, 23, 28, 28

Median Rank numbers, pick middle one What if data were…? 17, 19, 20, 23, 23, 28 Solution: add up two middle scores, divide by 2 (=21.5) Best measure in asymmetrical distribution (i.e. skewed), not sensitive to extreme scores Ex. 17, 19, 20, 23, 23, 428

Mean = X Add up the numbers and divide by the sample size (the number of numbers!) Try this one… 2,3,5,6,9 2+3+5+6+9 = 25 / 5 = 5 (Usually) best measure of the three –uses the most information (all values from distribution contribute)

Characteristics of the Mean Balance point Point around which deviations sum to zero Deviation = X – X For instance, if scores are 2,3,5,6,9 Mean is 5 Sum of deviations: (-3)+(-2)+0+1+4=0 ∑ (X – X) = 0

Characteristics of the Mean Affected by extreme scores Example 1 Scores 7, 11, 11, 14, 17 Mean = 12, Mode and Median = 11 Example 2 Scores 7, 11, 11, 14, 170 Mean = 42.6, Mode & Median = 11

Characteristics of the Mean Balance point Affected by extreme scores Appropriate for use with interval or ratio scales of measurement More stable than Median or Mode when multiple samples drawn from the same population Basis for inferential stats

Guidelines to Choose Measure of Central Tendency Mean is preferred because it is the basis of inferential statistics Median may be better for skewed data Distribution of wealth in the US – ex. annual household income in Washington state for 2000: mean=$76,818; median=$42,024 Mode to describe average of nominal data (eye color, hair color, etc…)

Normal Distribution Frequency, How often a score occurs Scores

MLB batting averages over 3-year span (min. 100 AB) Mean = 0.267 n = 1291

Normal Distribution Scores Mode “Normal” distribution indicates the data are perfectly symmetrical Median Mean Scores

Positively skewed distribution Mode Median Mean Scores

NFL Salaries 2011

Negatively skewed distribution Mode Median Mean Scores

Relationship among the MCT & shape of distribution

Alaska’s average elevation of 1900 feet is less than that of Kansas. Nothing in that average suggests the 16 highest mountains in the United States are in Alaska. Averages mislead, don’t they? Grab Bag, Pantagraph, 08/03/2000

Measures of dispersion or spread Variability Measures of dispersion or spread The only thing constant is variation.

the notions of variability Unusual Peculiar Strange Original Extreme Special Unlike Deviant Dissimilar different NY Times, 10/24/ 2010 Stories vs. Statistics By JOHN ALLEN PAULOS

Variability defined Measures of Central Tendency provide a summary level of the data Recognizes that scores vary across individual cases ie, the mean or median may not be an actual score in your distribution Variability quantifies the spread of performance How scores vary around mean/mode/median

To describe a distribution 1) Measure of Central Tendency Mean, Mode, Median 2) Measure of Variability Multiple measures Range, Interquartile range, Semi-Interquartile Range Standard Deviation

Range Range = Difference between low/high score # of hours spent watching TV/week 2, 5, 7, 7, 8, 8, 10, 12, 12, 15, 17, 20 Range = (Max - Min) Score 20 - 2 = 18 Very susceptible to outliers Doesn’t indicate anything about variability around the mean/central point

Semi-Interquartile range What is a quartile?? Divide sample into 4 parts of equal size Q1 , Q2 , Q3 = Quartile Points Interquartile Range = Q3 - Q1 Difference between highest and lowest quartile SIQR = IQR / 2 Related to the Median…prevents outliers from overly skewing measure For ordinal data or skewed interval/ratio

Quartiles based on miles walked/week BMD and walking Quartiles based on miles walked/week Krall et al, 1994, Walking is related to bone density and rates of bone loss. AJSM, 96:20-26

Notes: Skewed Distribution? 95th Percentile? 50th Percentile vs Median?

Standard Deviation Most commonly accepted measure of spread Variation itself is nature's only irreducible essence. Stephen Jay Gould Standard Deviation Most commonly accepted measure of spread Compute the deviations of all numbers from the mean Square and THEN sum each of the deviations Divide by the number of deviations Finally, take the square root

Standard Deviation Distribution = 1, 3, 5, 7 X = 16 /4 = 4 1) Compute Deviations = -3, -1, 1, 3 2) Square Deviations = 9, 1, 1, 9 3) Sum Deviations = 20 4) Divide by n= 20/4 = 5 5) Take square root = √5 = 2.2

Key points about SD SD small  data clustered round mean SD largedata scattered from the mean Affected by extreme scores (just like mean)…oftentimes called “outliers” Consistent (more stable) across samples from the same population Just like the mean - so it works well with inferential stats (where repeated samples are taken)

SD Example Three NFL quarterbacks with similar QB ratings in 2006: Matt Hasselbeck (SEA) = 76.0 Rex Grossman (CHI) = 73.9 Brett Favre (GB) = 72.7 Note: QB rating involves a complex formula accounting for passing attempts, completions, yards, touchdowns, and interceptions…100+ is considered outstanding & 70-80 is average All appear to have had very similar, somewhat mediocre seasons as QB’s

SD Example Let’s look at the SD of their game-by-game QB ratings: Matt Hasselbeck (SEA) = 29.97 Rex Grossman (CHI) = 47.60 Brett Favre (GB) = 27.81 Grossman had, by far, the most variability (i.e. inconsistency) in his game-by-game performances…is this good or bad?

Clinical Use of SD

SD and the normal curve The following concepts are critical to your understanding of how descriptive statistics works Remember – a “normal” curve is perfectly symmetrical. This is not typical, but usually data are almost normal…

SD and the normal curve X = 70 SD = 10 34.1% 34.1% About 68% of scores fall within 1 SD of mean X = 70 SD = 10 34.1% 34.1% 60 70 80

The standard deviation and the normal curve About 68% of scores fall between 60 and 70 X = 70 SD = 10 34% 34% 60 70 80

The standard deviation and the normal curve About 95% of scores fall within 2 SD of mean X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 50 60 70 80 90

The standard deviation and the normal curve About 95% of scores fall between 50 and 90 X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 50 60 70 80 90

The standard deviation and the normal curve About 99.7% of scores fall within 3 S.D. of the mean X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% 40 50 60 70 80 90 100

The standard deviation and the normal curve About 99.7% of scores fall between 40 and 100 X = 70 SD = 10 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% 40 50 60 70 80 90 100

What about = 70, SD = 5? What approximate percentage of scores fall between 65 & 75? …1SD below + 1SD above = 68% What range includes about 99.7% of all scores? …3SD below to 3SD above = 55 to 85

Interpreting The Normal Table Area under Normal Curve Specific SD values (z) include certain percentages of the scores Values of Special Interest 1.96 SD = 47.5% of scores (47.5 + 47.5 = 95%) 2.58 SD = 49.5% of scores (49.5 + 49.5 = 99%) ie, 95% of scores fall within 1.96 standard deviations of the mean (1.96 above and 1.96 below)

IQ X = 100 SD = 15 68% have an IQ between 85-115 55 70 85 100 115 130 34.1% 34.1% 13.6% 13.6% 2.3% 2.3% 55 70 85 100 115 130 145

MLB players’ batting averages over a 3-year span (min. 100 at bats) ~95% of players have an average between 0.196 and 0.337

Next Week… We will utilize our understanding of descriptive statistics concepts, including central tendency, variability, and the normal curve, to examine standardized scores Homework = Cronk 3.1 – 3.4 Bring calculator to class In-class activity 2…