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

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.

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

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?

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

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)

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…

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).

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

10 Stem and leaf plot: example data

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

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.

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

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

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)

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

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%

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)

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

20 Cumulative Frequency as a Percentage = = 10

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

Introduction to Probability

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.

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

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)

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 = (the RF of landing a “4”)  Relative Frequency is also the probability, so: p (throwing a 4) = p (not throwing a 4) = = 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).

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)

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

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

Let’s work on some exercises