Graphing and analysing the Normal distribution DS4CEC Distributions Graphing and analysing the Normal distribution

Histograms and Polygons Draw a histogram and a frequency polygon and curve for the data given. The frequency polygon can be drawn by plotting the points and joining them, but usually both the columns and line are drawn. The frequency curve usually gives an indication of shape.

Shapes of Distributions The shape of frequency curves may be described in terms of smoothness, symmetry and number of modes. The mode is the score with the highest frequency and has the highest value on the graph. Graph A is a smooth curve, Graph B is not smooth. Graph C is unimodal (has one mode) and graph D is bimodal (has two modes). Note: A curve with two distinct humps, even if the frequency at each hump is not the same, is called bimodal. If a curve has three or more modes it is sometimes described as being multimodal.

Skewness Graph E is symmetrical, Graph F is asymmetrical. Graphs that are not symmetrical are said to be skewed. If the longer tail of the graph is to the left, then the distribution is negatively skewed. This would occur, for example, if we graphed the distribution of the results of a very easy test. Most of the students would score high marks and only a few would score low marks. If the longer tail is to the right, then the distribution is positively skewed. This would occur for the distribution of the results of a very hard test. This curve is described as bell-shaped. It occurs for many naturally occurring characteristics, such as the number of tomatoes on a plant, the number of peas in a pod, the heights of a particular age group of females.

Other Distribution Shapes This is called a J-shaped distribution because of its similarity to the shape of this letter. As the value of the variable increases so does the frequency of that variable. For a reverse J-shaped distribution, the value of the variable decreases as the frequency of occurrence increases. A U-shaped distribution is U-shaped. A uniform distribution has no mode.

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Match each description with one of the curves shown below. a smooth, symmetrical, U-shaped b smooth, reverse J-shaped, mode at end of data c uniform distribution, no mode d smooth, unimodal, positively skewed e smooth, unimodal, negatively skewed f smooth, symmetrical, bell-shaped, unimodal g smooth, symmetrical, bimodal h smooth, J-shaped, mode at end of data i not smooth, asymmetrical, multimodal

The Normal Distribution A distribution of scores that produces a curve with this shape is known as a normal distribution. For a normal distribution: • The frequency curve is bell-shaped and symmetrical about the mean. • The mean, median and mode are equal. The size and shape of the bell are determined by the mean and standard deviation of the distribution. These three normal curves have the same mean but different standard deviations. These three normal curves have the same standard deviation but different means.

Standard Deviation If a normal distribution of scores has a large standard deviation, the scores are widely spread from the mean and the graph will be short and widely spread (below left). If the standard deviation, s, of the scores is small, most of the scores are close to the mean and the graph will be tall and narrow (below right).

Normal Distribution Properties

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