Know the characteristics of normal and skewed distributions (to include:…

The idea

A frequency distribution is the shape a set of scores makes when the scores run along the horizontal axis and the frequency up the vertical one. The bell shaped curve is symmetrical about its centre, so the mean, the median and the mode all sit together under the peak, the tails get closer to the axis without ever touching it, and roughly 68 percent of scores fall within one standard deviation of the mean with roughly 95 percent within two. Asymmetry pulls those three averages apart. A long tail of high scores drags the mean out towards the tail and leaves the mode under the hump, so reading from left to right the order becomes mode, median, mean, and a long tail of low scores reverses that order.

What earns the marks

• Describing the normal distribution precisely: symmetrical and bell shaped, with the mean, the median and the mode at the same central point and the tails approaching but never meeting the horizontal axis.
• Placing the averages correctly on a drawn curve, so on a positively skewed curve the mode is at the peak, the median comes next and the mean lies furthest into the long right hand tail.
• Naming the skew from the tail rather than from the hump, so a long tail stretching towards the high scores is a positive skew and a long tail towards the low scores is a negative skew.
• Explaining the shape from the study, for example that a test almost everybody failed produces a positive skew because most scores bunch at the low end and only a few stretch upwards.
• Saying what follows for reporting, namely that the median is the more representative average in a skewed distribution because the mean is dragged towards the extreme scores.

Common mistakes and how to put them right

• Labelling the skew by where most of the scores sit rather than by where the tail points, so a curve with the bulk of scores on the left is wrongly called negatively skewed.

• Putting the mean at the peak of a skewed curve, when the peak is the mode and the mean is the average that gets pulled out towards the tail.

• Writing that a normal distribution has the mean in the centre and stopping there, without adding that the median and the mode lie at the same point.

• Confusing skew with dispersion, and describing a wide flat curve as skewed when it is symmetrical but simply more spread out.

Exam tips

• These marks often arrive as a diagram to annotate, with arrows to label or a curve on which you must mark the position of the mean, the median and the mode, so practise the drawing and not only the wording.
• A stem that mentions a floor effect or a ceiling effect is telling you which skew to expect, because a task that was far too hard gives a positive skew and one that was far too easy gives a negative skew.
• If asked to explain why a researcher reported the median, tie the answer to the skew and to the outlying scores rather than saying that the median is easier to find.
• Where a curve is described as normal, you can use it: say that about 68 percent of participants scored within one standard deviation of the mean when the question asks you to interpret a value.