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.

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

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.

Ratio & 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.

Describing Data Measure of Central Tendency Measures of Spread Measures of Form

Measures of Central Tendency There are three commonly used measures of central tendency: the mode the median the arithmetic mean.

The Mode The mode is the score, or the score interval that occurs most frequently in the data.

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

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

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

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

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

Measures of Spread Distance based measures include: The range The interquartile range Centre based measures include: The variance The standard deviation.

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

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.

Variance The variance of a set of scores is defined as: As the spread in a set of scores increases so does the variance.

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.

Using Measures of Centre and Spread Nominal Mode Ordinal Median Range, Interquartile Range Interval/Ratio Mode, Median, Mean Range, Interquartile Range, Variance, Standard Deviation

Measures of Form The Normal Distribution Skew Kurtosis

The Normal Distribution

Skew Positive Skew Negative Skew

Kurtosis Leptokurtosis Platykurtosis