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PES 218 Final Exam
Analytic Methods - ANS>use data to draw conclusions and make decisions. However, the challenge of using these methods is that sometimes the messages of the data are not always clear. Discussion - examples? Last time - data set - area codes One distinguishing feature of analytic methods is that they recognize the inherent randomness of data and they are designed to extract useful information in the presence of this randomness. In analyzing sports data, results depend not only on the skill of the participants, but also on luck and randomness. Chapter 1 Review IMPORTANT - ANS>When using any type of statistical models to analyze data, we must keep in mind that they(statistical models) involve some idealization and simplification of a complicated physical relationship. This is a GOOD THING! Appropriate simplification is a crucial step in stripping away the randomness that often clouds our perceptions that drive sports results. Another role of statistical concepts in analytic methods is to give a framework for using probability to describe uncertainty (what does that mean?)
Given the random nature of the results of sports events, any conclusions we draw from them will naturally have some uncertainty related to them. Subject - ANS>an object on which data are collected. For sports data, the subjects are often players, teams, games, seasons, coaches. Variable - ANS>a characteristic of a subject that can be measured (like hits for a Major League Baseball player). The set of all variables for the subjects under consideration makes up the data to be analyzed. Statistics - ANS>a set of procedures and rules for reducing large masses of data to manageable proportions and for allowing us to draw conclusions from these data. Dr. Ballou's Definition of Statistics - ANS>a set of mathematical procedures for estimating what is probably going on in the world around us. This information is drawn from a population. Population - ANS>an entire collection of events in which you are interested. A collection of individuals exhibiting sets of shared traits, characteristics, and/or behaviors that emerge and develop under common, contextual (Ballou's 3/10 rule) influences.
Quantitative Data - ANS>A variable with a measurement scale consisting of numbers; responses are assigned a number value Qualitative Data - ANS>A variable with a measurement scale containing a set of code words as values (categories, spoken word, telling of an experience) Measurement Scale - ANS>the set of values a variable can take (discussion) Measurement scales can be sets of numbers (quantitative) or word based (qualitative) Scientific Quantitative Research - ANS>1. Define a population - who we want to make inferences about
- Traits, characteristics, behaviors that we want to learn about through some type of constructed research question
- Method of measurement - response format, how do we study the population?
- Sample - randomly selected, sample size = n (commit this to memory)
- Data - the set of numbers (quantitative) obtained by measuring the individuals in the sample
- Statistics - analyses (how do we analyze the collected data?)
- Draw conclusions - or make inferences related to what we think we should find
Discussion Continuous Variables - ANS>these are variables that can take on any value within a range of values Discrete Variables - ANS>these are variables that have a small set of values, only certain values can appear, often times can be part of a list of patterns Nominal Scale - ANS>values that are not quantities, but category labels; no mathematical properties (for example, assign 1 to blue, assign 2 to green, et al) Ordinal Scale - ANS>a nominal variable in which the possible values are ordered; not quantities, but labels for categories that can be ordered (class rank for example); minimal mathematical properties Interval Scale - ANS>are quantities, but NO absolute zero point; values are arbitrary (not based on reason); moderate level of mathematical properties; temperatures on a thermometer are interval scales; 0 degrees does not mean there is no temperature Ratio Scale - ANS>are quantities; do have an absolute 0 point and that point means nothing, or no value (discussion); 0 means 0; complex
Mode - ANS>the value with the highest frequency Mode values should always be kept separate - doesn't necessarily relate to some studies The mode can be computed for any type of variable, but for nominal values it is the only tool for central tendency There can be more than one mode if results display a bi or tri-modal data set Median - ANS>the value that is in the middle of an ordered data set Very simple formula to get it (n X 1/2); Determine the location/position of the median to get the value The median is the value that provides a summary toward a central tendency - the middle range If the data set is even you can take the average of the median scores on either side Mean - ANS>is the average of the data set; total the scores and divide by the number of the entire set (n); nearly every calculation in statistics/analytics is based off the mean; always use it as your starting point Most mathematically complex of the three central tendency measures Often labeled ̅ X Means are very sensitive to extreme scores (outliers) especially when there are fewer scores in the data set
Extremely useful tool in real life (test scores) Dispersion - ANS>the degree to which individual data points are distributed around the mean. Variance - ANS>The average of the squared differences from the Mean. Variance is based on the concept of deviation from the Mean and often labeled s² or σ². For any distribution of scores measured on a continuous scale, we can compute a Mean score, then measure the distance of each score from the Mean. Explaination of Variance - ANS>Measurements of the Variations in Data In all things, including sports, data have variations Understanding and accounting for variation is a central goal of analytic methods Variability - the deviation in the scores from individual to individual; there is useful information in the variables Dispersion - how are these scores dispersed or deviated from the central values? Mean, Median, Mode First measure of Dispersion is the Range, or the difference between the highest and lowest values (discussion/examples) Second measure of Dispersion is the Variance (more to follow)
IMPORTANT: Probability - ANS>- in the analytic view is simply the analysis of possible outcomes Example: Professor Amy likes candy. Her personal preference is Snickers, but on occasion she eats a Milky Way. Professor Amy keeps 100 mini candy bars in her desk. Of those 100, she has 85 Snickers (A) and 15 Milky Ways (B). When she is hungry she reaches in her desk drawer and grabs a candy bar at random. Because 85 candy bars are Snickers, and because she is grabbing at random, the probability of her drawing a Snickers is 85/ =. If an event can occur in A ways and can fail to occur in B ways, and if all possible outcomes are equally likely (each candy bar has an opportunity of being drawn), then the probability of its occurrence is: A/(A + B), and the probability of its failing to occur is B/(A + B). Probability - the relative frequency views probability in terms of past performance (very, very, very, very useful in sport probability); the text uses the definition "long run relative frequency" (Severini, 2015, p. 42). So we could define probability as the limit of the relative frequency of occurrences of the desired event that we expect as the number of draws increases.
Central Limit Theory - ANS>states our case that the more subjects in a data set, the closer we get to the true value. Rules of Probability - ANS>Probabilities all range from .00 to 1.00. If an event has a probability of .00 it is certain not to happen; if an event has a probability of 1.00 then it is certain to occur Two events are said to be independent when the occurrence or non- occurrence of one event has no effect on the occurrence or non- occurrence of the other. Two events are said to be mutually exclusive if the occurrence of one event precludes (is a factor in) the occurrence of the other. A set of events is said to be exhaustive if it includes all possible outcomes. Independent Events - ANS>are not affected by previous events. This is a very important idea. For example, a coin does not know if it came up heads before Each toss of a coin is a perfectly isolated event Mutually Exclusive Events - ANS>cannot occur at the same time For example, a coin cannot be heads and tails on the same flip A sport example:
This is exactly what you have in the NFL kicker example. Each of you are comparing your individual kicker to the group in 10 categories. Based on your Z score calculation, students will go to the provided Z score table and identify where the Z score lies. The Z score value can be either positive or negative, which simply means that the score is either above or below the group mean. With that, we want to know how far above or below the group mean.tell us whether a particular score is equal to the mean, below the mean, or above the mean of a bunch of scores. Z Scores - tell us how far a particular score is away from the mean (close, far away). Z Scores - tell us how typical a particular score is within a bunch of scores. If data are normally distributed, approximately 95% of the data should have a Z Score between -2 and +2 (SD's). Z Scores that do not fall within this range may be less typical of the data in a bunch of scores. Null hypothesis (Ho) - ANS>The null hypothesis states that a population parameter is equal to a value (usually the mean). The null hypothesis is often an initial claim that researchers specify using previous research or knowledge. The Null Hypothesis (Ho) reflects that there will be no observed effect for our experiment. In other words, the Ho is the hypothesis of NO difference or of NO relationship. The Null Hypothesis is what we attempt to find evidence against in our experiment (or test). While conducting our experiment, we will find evidence for two results:
Reject the Ho (this does not mean we adopt the H:a) Fail to reject the Ho (this does not mean we reject the H:a) If we fail to reject the Ho, we can say that Xob = ¯Xbar If we reject the Ho, we can say that Xob ≠ ¯Xbar Alternative Hypothesis (Ha) - ANS>The alternative hypothesis states that the population parameter is different than the value of the population parameter in the null hypothesis. The alternative hypothesis is what you might believe to be true or hope to prove true. The Alternative Hypothesis (Ha) reflects that there will be an observed effect for our experiment. We adopt Ha when Ho is rejected. The Ha is what we are attempting to demonstrate in an indirect way by the use of our hypothesis test: If the Ho is rejected, then we adopt the Ha If the Ho fails to be rejected, then we do not adopt the H:a If we reject the Ho, then we can say that Xob ≠ ¯Xbar Hypotheses Statements - ANS>Ho - Based on statistical evidence, _____________ will = ________________________ at
______. Ha - Based on statistical evidence, _____________________ will ≠ ____________________ at
Hypothesis for Pearson's R - ANS>Ho: says there is no relationship between the two variables Ha: says there is a relationship between the two variables Pearson's R IMPORTANT FACTS - ANS>Pearson's r measures the strength of the linear relationship between two variables. Pearson's r is always between -1 and 1. A Pearson's r measurement of 1 means that there is a perfect POSITIVE linear relationship between two variables (text page 108). This means that if X increases, Y increases in exactly the same way. A Pearson's r measurement of -1 means that there is a perfect NEGATIVE linear relationship between two variables (text page 108). A Pearson's r measurement of 0 means that there is no positive or negative linear relationship between two variables (text page 108). Calculating Pearson's R - ANS> Pearson's R correlation coefficient - ANS>Where r is the correlation value between the two variables n is the sample size ∑XY is the sum of the cross products (X times Y) ∑X is the sum of the X's ∑Y is the sum of the Y's
∑X² is the sum of the X's squared (each X value squared, then summed) (∑X)² is the sum of the X's, quantity squared ∑Y² is the sum of the Y's squared (each Y value squared, then summed) (∑Y)² is the sum of the Y's, quantity squared
