C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
1
Why do we do statistics?
• To Make Inferences from a Small number of cases to a Large number of cases
• This means that we have to collect data.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
2
Kinds of Data
• There are four basic kinds of scores that can be collected.• Nominal• Ordinal• Interval• Ratio
• They are known as different types of scales
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
3
Nominal and Ordinal Scales
• Nominal/Categorical Scales• Here numbers are used to divide different behaviours into
different classes without implying that the different classes are numerically related to each other.
• Ordinal/Ranking Scales• Here numbers are used to indicate a relative position of
something in a list. Although we can decide what appears where on a list, in other words the order of the list, we cannot conclude that each item on the list appear at equal intervals.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
4
Ratio & Interval Scales
• Interval Scales• With interval scales we can take a step further than we
could with ordinal scales. Interval scales mean that each increase in unit value is one more than the previous value.
• Ratio Scales• Although interval scales are useful it is sometimes
important to use ratio scales. This is where there are equal intervals with an absolute zero value. Making statements about "A has twice as much x as B" sensible.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
5
Describing Data
• Measure of Central Tendency• Measures of Spread• Measures of Form
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
6
Measures of Central Tendency
• There are three commonly used measures of central tendency: • the mode • the median • the arithmetic mean.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
7
The Mode
• The mode is the score, or the score interval that occurs most frequently in the data.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
8
The Median
• The median is the score that's at the centre of a distribution of data in the sense that one half of the data values are less than the median, and one half are greater than the median
• So with the following data we obtain.• 17 23 45 46 83 84 96.
• The number that sits exactly in the middle is 46.• So half the numbers (17, 23, 45) are below the median and
half the numbers (83 84 96) are above the median
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
9
The Arithmetic Mean
• The arithmetic mean is nothing more than all of the scores added up and divided by the number of scores in the data set.
• The formula is normally given as:
• So the mean values of the numbers• 17 23 45 46 83 84 96
• comes to• 17+23+45+46+83+84+96/7 = 56.8
x x
N
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
10
Properties of the Measures of Centre
• All the measure of the centre have two important properties.• Adding or subtracting a constant does the same thing to
the measure of centre• Multiplying or dividing by a constant does the same thing
to the measure of centre
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
11
Properties of the Measures of Centre
• Of the measures the mode is the only one that is appropriate to use with category or nominal data.
• The median is especially useful for ordinal data but is occasionally used with interval and ratio data as well.
• The arithmetic mean requires arithmetic operations on intervals between values. For that reason it is only meaningfully used with data that have equal intervals, i.e. data collected using interval and ratio scales
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
12
Why Look at Measures of Spread and Form?
• More scores to the right of your score though your score and the class means are the same
• Even more scores to the right of your score though your score and the class means are the same
6253 6253
62536253
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
13
Measures of Spread
• Distance based measures include:• The range• The interquartile range
• Centre based measures include:• The variance• The standard deviation.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
14
Distance Based Measures of Spread
• Range = Largest Value - Smallest Value• Interquartile= Largest Value - Smallest Value after the top
and bottom 25% of the distribution is removed
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
15
Centre Based Measures of Spread
• Several statistics use the centre of the data as a point of reference and reflect how data clustered around it.
• Of these, the variance and the standard deviation are the most widely used.
• Both of which are based on the arithmetic mean.
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
16
Variance
• The variance of a set of scores is defined as:
• As the spread in a set of scores increases so does the variance.
s2 x x 2N
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
17
Standard deviation
• The standard deviation of a set of scores is defined as:
• As the spread in a set of scores increases so does the standard deviation.
• The standard deviation is on the same scale as the original set of scores.
s x x 2N
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
18
Using Measures of Centre and Spread
• Nominal• Mode
• Ordinal• Median• Range, Interquartile Range
• Interval/Ratio• Mode, Median, Mean• Range, Interquartile Range, Variance, Standard Deviation
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
19
Measures of Form
• The Normal Distribution• Skew• Kurtosis
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
20
The Normal Distribution
Mean
ModeMedian
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
21
Skew
• Positive Skew
• Negative Skew
Mean
ModeMedian
Mean
ModeMedian
C82MCP Diploma Statistics
School of PsychologyUniversity of Nottingham
22
Kurtosis
• Leptokurtosis
• Platykurtosis