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1 Practical Psychology 1 Week 5 Relative frequency, introduction to probability

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2 Example: We want to order shoes for 12 girls Measure the shoe-sizes of 12 girls in Greece N = 12 (sizes: 39, 41, 40, 37, etc). If the mean shoe-size turns out to be 39.25, does this mean we should order 12 pairs of size 39.6? In some situations, calculating the mean (as a measure of averageness) would not be useful.

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3 Need to answer 3 Questions: What scores did we obtain? (i.e. what was each person’s shoe size?) How many times did each score occur? (i.e. how many pairs of each size do we need to order?) How can we show this information? (representation of information) Thus, need to consider Frequency Distribution

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4 Some definitions… Frequency (f) – the number of times a score occurs in a set of data Frequency Distribution or histogram – a graph showing how many times each score occurs. What scores did we obtain? How many times did each score occur?

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5 Constructing Frequency Distributions: Tables Ungrouped Frequency distributions Grouped Frequency distributions Histograms Stem-and-leaf plots Bar charts

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6 TABLE: Ungrouped Frequency Distribution Score = N of relatives Freq = N of people with that particular number of relatives N = 10 (i.e. entire sample size)

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7 Example: IQ We could measure each person’s IQ score out of 100. This data could be represented as an ungrouped frequency distribution (like the previous slide) OR…

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8 Grouped Frequency Distribution Note that the class intervals are equal Class interval = 10 Need to select intervals carefully (must not be too narrow or too wide).

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9 Visually depicting the FD (relatives data): Histogram f goes on Y-axis Interval goes on X-axis relatives

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10 Stem and leaf plot: example data

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11 Visually depicting the FD: Stem & Leaf plot Leaf shows the final digits of the score Stem shows leading digits

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12 Remember! The previous examples work best with interval (or ratio) data Note that in a histogram, the X-axis is measuring a continuous variable, so the bars do touch.

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13 Frequency Distributions with nominal (i.e. categorical) data Sample data (N=10): 10 voters interviewed n (labour) = 6 n (con) = 2 n (lib-dem) = 2 To depict this data, can draw a bar chart

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14 Bar-chart Bars do NOT touch, as measuring categorical variable

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15 Relative Frequency and the Normal Curve Relative frequency is the proportion of the time that a score occurs in a data set. Indicated as a fraction between 0 and 1 (i.e. 0.1, 0.2, 0.3, 0.4,…1)

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16 Relative Frequency E.g. 1, 2, 2, 2, 3, 4, 4, 5 (N=8) RF of 2 is 3/8 = 0.375 = 0.38 (note: round off to 2 dp). Therefore, the score of 2 has a RF of 0.38 in the above data set.

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17 Relative Frequency as a percentage E.g. 1, 2, 2, 2, 3, 4, 4, 5 (N=8) RF of 2 is 3/8 = 0.38*100 = 38%

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18 RF and the Normal Curve Area under the curve is 100% of the sample So a proportion of the area under the curve corresponds to a proportion of the scores (i.e. the relative frequency)

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19 Examples If a score occurs 32% of the time, its relative frequency is 0.32 If a score’s relative frequency is 0.46, it occurs 46% of the time Scores that occupy 0.2 (20/100) of the area under the curve have a relative frequency of 0.2

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20 Cumulative Frequency as a Percentage 10 + 45 = 55 2.5 + 7.5 = 10

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21 Suggested Reading FIELD, A. (2009). Discovering Statistics using SPSS (3 rd ed.). London: Sage. pp. 18-20. LANGDRIDGE, D. (2004). Introduction to research methods and data analysis in Psychology. Harlow: Pearson – Prentice Hall. pp. 123-127.

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Introduction to Probability

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23 What is probability? E.g. coin: p (getting Heads) = 1 in 2 or 0.5 or 50% Probability can be expressed as a ratio, fraction, or percentage. Probability (p) describes random or chance events refers to how likely a particular outcome is. Event must be random (i.e. not rigged), so outcome be determined by luck.

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24 Probability of events occurring is measured on a scale from 0 (not possible) to 1 (must happen). 0 1

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25 Probability and Relative Frequency If an event occurs frequently over a period of time, high probability of occurring. If an event occurs infrequently over a period of time, low probability of occurring. This judgment is the event’s relative frequency, which is equal to it’s probability (see next slide for example)

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26 Probability and Relative Frequency RF of “4” occurring on a throw of a die is 1/6: 1 = frequency of event, 6 = total number of possible outcomes 1/6 = 0.167 (the RF of landing a “4”) Relative Frequency is also the probability, so: p (throwing a 4) = 0.167 p (not throwing a 4) = 1 - 0.167 = 0.833 Both probabilities should add up to 1. In research (or in life!), probabilities are often somewhere in between 0 and 1 - nothing is absolutely uncertain (or certain).

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27 Probability Distributions A probability distribution indicates the probability of all possible outcomes. Very simple To create a true probability distribution, need to observe the entire population. However, this isn’t always possible, so the probability distribution may be based on observations from a sample. Score on DieP (getting score on die) 10.167 2 3 4 5 6

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28 Creating a probability distribution from a sample, based on actual observations The arrival of a bus is observed for 21 days. Days on time = 7 Days late = 14 The Probability distribution on the basis of above sample is: p (on time) = 7/21 = 0.33 p (late) = 14/21 = 0.67

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Quick quiz… Relative frequency is indicated as a fraction between ……… and …………. Relative Frequency is also the ……………………… Probability refers to how …………. a particular outcome is. If an event occurs frequently over a period of time, it has a ……… probability of occurring. In a …………………, the X-axis is measuring a continuous variable, so the bars do touch

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Let’s work on some exercises

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1 Outline 1. Why do we need statistics? 2. Descriptive statistics 3. Inferential statistics 4. Measurement scales 5. Frequency distributions 6. Z scores.

1 Outline 1. Why do we need statistics? 2. Descriptive statistics 3. Inferential statistics 4. Measurement scales 5. Frequency distributions 6. Z scores.

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