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Descriptive Statistics: Central Tendency

Descriptive Statistics: Central Tendency. Lesson 4. Psychology & Statistics. Goals of Psychology Describe, predict, influence behavior & cognitive processes Role of statistics Descriptive statistics Describe, organize & summarize data Efficient communication Inferential statistics

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Descriptive Statistics: Central Tendency

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  1. Descriptive Statistics:Central Tendency Lesson 4

  2. Psychology & Statistics • Goals of Psychology • Describe, predict, influence behavior & cognitive processes • Role of statistics • Descriptive statistics • Describe, organize & summarize data • Efficient communication • Inferential statistics • Draw conclusions about data • Aid decision making ~

  3. Organizing Data • Describing distribution of variables • enumeration: list raw data • Frequency distributions • organize  tables or graphs • highlight important characteristics • range, most frequent value ~

  4. Distributions as Tables • f = Frequency • # of times a value of variable occurs • Sf = n • calculate proportions & percentages • Frequency distribution tables • ordered list of all values of variable & their frequencies • logical order (usually descending) ~

  5. Frequency Distribution Table X f 19 1 18 2 16 3 15 3 14 5 13 2 12 6 11 7 10 3 9 6 8 5 7 3 6 2 5 2 8 9 7 8 16 7 10 11 16 14 12 13 12 13 12 14 8 9 15 12 18 14 14 12 8 11 11 9 9 18 10 14 16 6 11 15 9 19 12 5 15 11 7 9 5 6 8 10 11 11 50 Enumeration # of presentations to be able to recall 100% • Sf = n • calculate proportions & percentages

  6. Grouped Frequency Distribution • Group by class intervals • report frequencyfor each interval • Lose information: no exact values • General rules • each interval same width • consecutive & do not overlap ~

  7. Frequency Distribution Table X f 19 1 18 2 16 3 15 3 14 5 13 2 12 6 11 7 10 3 9 6 8 5 7 3 6 2 5 2 8 9 7 8 16 7 10 11 16 14 12 13 12 13 12 14 8 9 15 12 18 14 14 12 8 11 11 9 9 18 10 14 16 6 11 15 9 19 12 5 15 11 7 9 5 6 8 10 11 11 50 Enumeration # of presentations to be able to recall 100% • Sf = n • calculate proportions & percentages

  8. Distributions as graphs • Summarizes data • focus on clear communication • Bar Graphs • nominal or ordinal data • discrete variables • Histograms & Frequency Polygons • Interval/ratio data • continuous & discrete variables • Relative frequency distributions • Y axis = proportions • Large data sets ~

  9. 18 18 14 14 10 10 f f 6 6 2 2 Rep Dem Ind A B C D F Exam Grades Political affiliation Bar Graphs Nominal Ordinal

  10. 18 14 10 f 6 2 21 5 13 15 19 7 9 17 11 19-20 5-6 7-8 9-10 17-18 11-12 13-14 15-16 # of presentations Histograms • X-axis • Class intervals of variables • Y-axis • Frequencies vertical bars ~

  11. 18 14 10 Relative Frequency 6 2 f 21 5 13 15 19 7 9 17 11 # of presentations # of presentations Frequency polygons • Frequency represented as points • Contains same info as histogram ~ f

  12. Distributions: 3 useful features • Summarizes important characteristics of data 1. What is shape of the distribution? 2. Where is middle of distribution? 3. How wide is distribution?

  13. Shapes of distributions • Unimodal distribution • single value is most frequent • Bimodal (or multimodal ) • 2 most frequently occurring values • May indicate relevant subgroups ~ f X f X

  14. f f f -4 -2 0 +2 +4 -4 -2 0 +2 +4 X X Symmetry of distributions • Symmetric • if right side mirror-image of left • Skewed - asymmetric • a few extreme values • Positively skewed: right tail longer • Negatively skewed: left tail longer ~

  15. f The Normal Distribution • Bell-shaped • 3 characteristics • Unimodal • symmetric • asymptotic • Many naturally-occurring variables approximately normally distributed • Makes statistics useful ~

  16. Central Tendency • Describes most typical values • Depends on level of measurement • Mode (all levels) • Most frequently occurring value • Median (only ordinal & interval/ratio) • value where ½ observations above & ½ below • Mean (only interval/ratio) • Arithmetic average ~

  17. 18 18 14 14 f 10 f 10 6 6 2 2 A B C D F Rep Dem Ind exam grades Political affiliation 18 18 14 14 f 10 f 10 6 6 2 2 21 5 13 15 19 7 9 17 11 21 5 13 15 19 7 9 17 11 # of presentations # of presentations Mode • Most frequently occurring value ~

  18. 10 20 30 40 50 60 70 80 90 10 20 30 40 50 60 70 80 90 Median • Midpoint of a data set • values ½ smaller, ½ larger ~

  19. Finding the Median 1. List all values from largest  smallest if f=3, then list 3 times 2. Odd # entries median = middle value 3. Even # entries = half way b/n middle 2 values ~

  20. Mean • Summarizes quantitative data • May not be actual value in data set • Introduces error • Most commonly used • Computing the mean Sum of all observations Mean = Number of observations

  21. Statistical Notation • Formula for mean: • Σ: summate • add all that follows • X: observation • value of an observation • N: number of observations • Or data points ~

  22. Populations & Samples: Notation • Different symbols • Often different formulas for calculation • Population: Greek letters • Population mean = μ • Sample: Roman letters • Sample mean = • APA style: M ~

  23. Populations & Samples • Population • Parameter • Exact value • Population mean = μ • Sample • Statistic • estimate of parameter • introduces error • Sample mean = ~

  24. Formulas for Mean • Population mean • Parameter • Sample mean • Statistic • Estimate / error • Sometimes n used for sample ~

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