Normal or Gaussian Distribution

Characteristics of Normal Distribution 

– Symmetrical about the mean

– Tails should meet the axis at infinity

– Bell-shaped distribution

– Mean = mode = median

– The area under the curve is 1 standard deviation away from the mean and makes up 68% of the entire distribution under the curve (This means that if you randomly select a point under the curve, there is a 68% chance it will fall one standard deviation from the mean)

– The area under the curve 1.96 SD (round to two) away from the mean makes up 95% of the entire distribution under the curve (This means that if you randomly select a point under the curve, there is a 95% chance it will fall 2 standard deviations from the mean)

– The sample mean = mean of the population

– The standard deviation of the mean distribution or standard error = (SD of the population)/(square root of the number of scores)

– The standard error indicates the degree to which sample means deviate from the mean

– The sample mean distribution converges to normal distribution as the size of the sample increases

– The bell-shaped curve can also be reflected in the lay-out of a histrogram

curve1

Here the SD is 15 units

curve2

Questions Dealing with Standard Deviation

Question: Assume the standard deviation is 10 and the mean score is 100. If you randomly select any point 1 standard deviation from the mean, what would be your range?

Answer: The range would be between 90 and 110. As one standard deviation is 10 units left or right. You could also say that you have a 68% chance of randomly picking a score between 90 and 110 on the this graph.

Question: Assume the standard deviation is 10 and the mean score is 100. If you randomly select any point 2 standard deviations from the mean, what would your range be?

Answer: The range would be between 80 and 120. As one standard deviation is 10 units left or right, 2 standard deviations would be 20 units left or right. You could also say that you have a 95% of randomly picking a score between 80 and 120 on this graph.

N.B: 95% is the commonly accepted probability, which is the alpha level or confidence level in psychological studies for rejecting the null hypothesis is p<0.05.

The z-Score 

It is possible to convert all normal distributions to the standard normal distribution.

For a standard normal distribution the mean has to equal 0 and the SD has to equal 1.

You can find the z-score by subtracting the mean from each data point, and then dividing the this zero-meaned data by the standard deviation.

If your final data point is +1, this point is one standard deviation above the mean. If your final data point is -3, this point is 3 standard deviations below the mean. The z-score is particularly useful for comparing data across different situations.

Error Bar Charts

Error bar charts are away of representing the confidence interval. Error bars display your mean means as a point on a chart and a vertical line through the mean point that represents the confidence interval. The longer the line, the longer the confidence interval. Error bar charts can also be used to see if two population means differ from each other by comparing confidence interval. If the confidence intervals do not overlap we can be 95% confident that both population means fall within the intervals indicated and therefore do not overlap.

curve3

Bibliography

ZHENG, Y. (2013). Referencing and citation – Harvard style, from PSY104 Methods and Reasoning for Psychologists. University of Sheffield, Richard Roberts Building on 11th February. Available from: Blackboard.
[Accessed 4/02/13].