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Lecture 6: Descriptive Statistics: Probability, Distribution, Univariate Data

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2 Agenda Wrap-up of experimental methods Intro to probability Examining data through univariate statistics

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3 Generalizability (external validity) in Experiments Threats to external validity always involve an interaction of the treatment group with some other factor. Threats usually fall into 3 types: Setting Setting Population Population History History

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4 Three threats to generalizability in experiments Setting Physical and social context of the experiment Physical and social context of the experimentPopulation Is there something specific about the sample that interacts with the treatment? Is there something specific about the sample that interacts with the treatment?History Is there something about the time that interacts with the treatment? Is there something about the time that interacts with the treatment?

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5 Why Generalizability is not always a problem Experiments often are trying to isolate specific causes and effects in controlled settings. Thus, they may not even be claiming to be generalizable to specific settings. Experimental findings can provide theoretical basis for real-world tests. It is often a balancing act for research: true causation versus large-scale associational and comparative testing.

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6 Considerations before using experiments Cost and Effort Is the effort worth it to test the concepts you are interested in? Is the effort worth it to test the concepts you are interested in? Manipulation and Control Will you actually be able to manipulate the key concept(s)? Will you actually be able to manipulate the key concept(s)? Importance of Generalizability Are you testing theory, or trying to establish a real- world test? Are you testing theory, or trying to establish a real- world test?

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8 Probability Are the things that we observe different from what would be expected by chance? Coin Example

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9 Probability Concepts Basic Concepts in Probability Basic Probability Rules Special Types of Probability Joint Probabilities Joint Probabilities Probabilities of Unions of Events Probabilities of Unions of Events Conditional Probabilities Conditional Probabilities

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10 Basic Concepts in Elementary Probability Random Selection Every possibility has equal chance of being chosen. Every possibility has equal chance of being chosen.Independence The probability of a response on one trial does not depend on the outcome of any other trials. The probability of a response on one trial does not depend on the outcome of any other trials. Elementary Event Possible outcomes of a probability experiment Possible outcomes of a probability experiment E.g., each coin toss E.g., each coin toss Sample Space The complete set of elementary events The complete set of elementary events E.g., all coin tosses E.g., all coin tosses

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11 Mutually exclusive, exhaustive, events Mutually exclusive events Two or more events that cannot occur at the same time. Two or more events that cannot occur at the same time. Exhaustive events A set of events that accounts for all of the elementary events in the sample space. A set of events that accounts for all of the elementary events in the sample space.

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12 Basic rules of probability Multiplication Rule For independent events, we can multiply the probabilities together to get the probability for all of the events occurring. For independent events, we can multiply the probabilities together to get the probability for all of the events occurring. Example: Probability of rolling a die and getting 6 on both rolls. But what happens if the events are not independent? Example: probability of selecting a club from a deck of cards, then selecting another club (without replacement)? Example: probability of selecting a club from a deck of cards, then selecting another club (without replacement)?

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13 Multiplication Rule when two or more events will happen at the same time, and the events are independent, then the special rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) x P(Y) when two or more events will happen at the same time, and the events are dependent, then the general rule of multiplication law is used to find the joint probability: P(X and Y) = P(X) x P(Y|X)

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14 Basic rules of probability (continued) The addition rule For independent events, we can add the probabilities to get the probability of either event occurring. For independent events, we can add the probabilities to get the probability of either event occurring. Example: Rolling die and getting a 4 or a 6. Again, what happens if the events are not independent (in this case, mutually exclusive)?

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15 Addition Rule When two or more events will happen at the same time, and the events are mutually exclusive, then: P(X or Y) = P(X) + P(Y) When two or more events will happen at the same time, and the events are not mutually exclusive, then: P(X or Y) = P(X) + P(Y) - P(X and Y) For example, what is the probability that a card chosen at random from a deck of cards will either be a king or a heart? P(King or Heart) = P(X or Y) = 4/52 + 13/52 - 1/52 = 30.77%

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16 Special Types of Probability Joint Probabilities Probabilities of Unions of Events Conditional Probabilities

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17 Example: Female and Download Music from Class Survey

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18 Joint Probabilities Probability of obtaining a particular combination of events. E.g., probability of flipping a coin twice and getting heads both times. Just use multiplication rule! E.g., probability of flipping a coin twice and getting heads both times. Just use multiplication rule! P (A and B) = n(A and B) / n (S) What about non-independent events? What about non-independent events? E.g., probability of a given respondent in class survey being female and having downloaded music before. P (A and B) = p(A|B) x p(B) (9/11) (.579) =.474 (9/11) (.579) =.474

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19 Union Probabilities A union of two elementary events consists of all the elementary events belonging to either of them. Examples: Probability of flipping a coin and it being heads or tails. (mutually exclusive union) Probability of flipping a coin and it being heads or tails. (mutually exclusive union) Non-independent events: Probability of being a female or having downloaded music before. Non-independent events: Probability of being a female or having downloaded music before. p(E1) + p(E2) – p(E1 and E2) (.579) + (.842) – (.474) =.947 (.579) + (.842) – (.474) =.947

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20 Conditional Probability Probability of an event occurring given that another event has occurred. Example: probability of an outcome, given that something else has occurred. Example: probability of an outcome, given that something else has occurred. 3 Doors Problem 3 Doors Problem

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21 Probability and Statistics Statistics deal with what we observe and how it compares to what might be expected by chance. A set of probabilities corresponding to each possible value of some variable, X, creates a probability distribution Common examples include normal (Gaussian), Poisson, Exponential, Binomial, etc Common examples include normal (Gaussian), Poisson, Exponential, Binomial, etc

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22 The Normal Curve

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24 For now, we will just deal with describing or characterizing the distribution of a single variable

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25 Describing Simple Distributions of Data Central Tendency Some way of “typifying” a distribution of values, scores, etc. Some way of “typifying” a distribution of values, scores, etc. Mean (sum of scores divided by number of scores) Median (middle score, as found by rank) Mode (most common value from set of values) In a normal distribution, all 3 measures are equal. In a normal distribution, all 3 measures are equal. Example: Class stats knowledge Example: Class stats knowledge

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26 Dispersion Range Difference between highest value and the lowest value. Difference between highest value and the lowest value. Standard Deviation A statistic that describes how tightly the values are clustered around the mean. A statistic that describes how tightly the values are clustered around the mean.Variance A measure of how much spread a distribution has. A measure of how much spread a distribution has. Computed as the average squared deviation of each value from its mean Computed as the average squared deviation of each value from its mean

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27 Properties of Standard Deviation Variance is just the square of the S.D. If a constant is added to all scores, it has no impact on S.D. If a constant is multiplied to all scores, it will affect the dispersion (S.D. and variance) S = standard deviation X = individual score M = mean of all scores n = sample size (number of scores)

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29 Common Data Representations Histograms Simple graphs of the frequency of groups of scores. Simple graphs of the frequency of groups of scores. Stem-and-Leaf Displays Another way of displaying dispersion, particularly useful when you do not have large amounts of data. Another way of displaying dispersion, particularly useful when you do not have large amounts of data. Box Plots Yet another way of displaying dispersion. Boxes show 75 th and 25 th percentile range, line within box shows median, and “whiskers” show the range of values (min and max) Yet another way of displaying dispersion. Boxes show 75 th and 25 th percentile range, line within box shows median, and “whiskers” show the range of values (min and max)

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