Data Analysis: Descriptive Statistics

What is descriptive statistics?

The use of graphs, tables and summary statistics to identify trends and analyse a set of data.

What is measures of central tendency?

Any measure which calculates an average value within a set of data.

• Gives us information about the most typical values in a set of data

• Mean, median and mode

What is the mean?

Total of all values in a set of data is divided by the number of values.

• E.g. values: 5,7,7,9,10,11,12,14,15,17 - added together to get the total 107, then divided by the amount of values (10), giving a mean value of 10.7

What is the median?

The middle value in a set of data when scores are arranged from lowers to highest.

• In an odd number of scores, the median is easily identified

• In an even number of scores, the median is halfway between the two middle scores

What is the mode?

The most frequently occurring value in a set of data.

• In some data sets, there may be two modes (bi-modal) or no mode if all the scores are different

What are the strengths of the mean?

• Most sensitive measure of central tendency as it takes all scores in the data set into account

• It’s more likely (than other measures of central tendency) to provide a representative score, meaning it’s the most reliable measure of central tendency

What are the limitations of the mean?

• Sensitive to extreme scores (outliers) so it can only be used when the scores are reasonably close - not suitable for some data sets

• The mean score may not be represented in the data set itself, the mean may not appear in the original data set itself

What are the strengths of the median?

• Not affected by extreme scores, can be used on data sets with anomalous (irregular) scores

• Best measure of central tendency when dealing with qualitative data, where ranking of categories or themes is used instead of measurement or counting

What are the limitations of the median?

• Doesn’t necessarily represent a typical average as it doesn’t include all of the data in its calculation - it doesn’t account for extreme scores which makes it less reliable than the mean

• Arranging the data in ascending or descending order is time consuming, making the median more problematic in dealing with large data sets

What are the strengths of the mode?

• Not affected by extreme values

• Often useful for the analysis of qualitative data, which may require frequencies of themes to be analysed

What are the limitations of the mode?

• A data set may include two modes which blurs the meaning of the data, making it difficult for the researcher to form conclusions - least reliable of the measures of central tendency

• Likely to be of little use on small data sets as it may provide an unrepresentative central measure e.g. a data set may include a mode of 73 when in fact the average score in that set is 55 - lacks validity

What is measures of dispersion?

The general term for any measure that calculates the variation in a set of data.

What is a data set with low dispersion?

• Scores are close together and near the mean

• Indicates little variation between participants

• Data is more consistent/reliable

- E.g. Test scores of 68, 70, 69, 71

What is a data set with high dispersion?

• Scores are spread out and far from the mean

• Indicates large variation between participants

• Data is less consistent

- E.g. Test scores of 30, 55, 70, 90

What is the range?

The difference between the lowest and the highest scores in a data set. Calculated by subtracting the lowest value from the highest value in the data set.

• E.g. Values: 4,4,6,7,9,9 - subtract the lowest number (4) from the highest number (9) so the range is 9

What is standard deviation?

Calculates how a set of scores deviates from the mean. It provides insight into how clustered or spread out the scores are from the mean.

What is low standard deviation?

Indicates that the scores are clustered tightly around the mean which indicates the reliability of the data set.

What is high standard deviation?

Indicates that the scores are more spread out from the mean which indicates lower reliability.

What are the strengths of the range?

• Provides a broad overview of the data which can be useful for some research purposes

• Simple and easy to calculate

What are the limitations of the range?

• Provides no information as to all of the other scores in the data set, meaning it lacks validity as it doesn’t indicate the degree of variation from the mean

• Not very stable or representative, can vary from one sample to another as sample size increases

What are the strengths of standard deviation?

• Provides information as to how the scores are distributed across a data set, it can indicate to what extent the data is reliable and consistent

• More sensitive than the range as it uses all the scores in the data set, it’s a more valid representation of the data set

What are the limitations of standard deviation?

• It’s onerous (inconvenient/troublesome) and time-consuming to calculate, however a statistical calculator or online tool can calculate the standard deviation much more quickly than in the past

• Can be skewed (biased/distorted) by extreme outliers, these may inflate or depress the standard deviation, giving a misleading representation of the spread of values in the data set

How do you calculate the standard deviation?

1. Calculate the mean 

2. Subtract the mean from each score in the data set

3. Square the scores which have just been calculated at step 2

4. Add all of the squared scores together 

5. Divide the total squared score by the number of scores minus 1

6. Work out the square root of the variance (using a calculator)