The measure of variance most often displayed with a mean to give an overall picture of the data is:

Summarising the dataset can help us understand the data, especially when the dataset is large. As discussed in the Measures of Central Tendency page, the mode, median, and mean summarise the data into a single value that is typical or representative of all the values in the dataset, but this is only part of the 'picture' that summarises a dataset. Measures of spread summarise the data in a way that shows how scattered the values are and how much they differ from the mean value.

4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11

If we just looked at the measures of central tendency, we may assume that the datasets are the same.

However, if we look at the spread of the values in the following graph, we can see that Dataset B is more dispersed than Dataset A. Used together, the measures of central tendency and measures of spread help us to better understand the data

What does each measure of spread tell us?

The range is the difference between the smallest value and the largest value in a dataset.

4, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8

1, 2, 3, 4, 5, 6, 6, 7, 8, 9, 10, 11

On a number line, you can see that the range of values for Dataset B is larger than Dataset A.

Quartiles divide an ordered dataset into four equal parts, and refer to the values of the point between the quarters. A dataset may also be divided into quintiles (five equal parts) or deciles (ten equal parts).

The lower quartile (Q1) is the point between the lowest 25% of values and the highest 75% of values. It is also called the 25th percentile.

The second quartile (Q2) is the middle of the data set. It is also called the 50th percentile, or the median.

The upper quartile (Q3) is the point between the lowest 75% and highest 25% of values. It is also called the 75th percentile.

As the quartile point falls between two values, the mean (average) of those values is the quartile value:
Q1 = (5+5) / 2 = 5
Q2 = (6+6) / 2 = 6
Q3 = (7+7) / 2 = 7

As the quartile point falls between two values, the mean (average) of those values is the quartile value:
Q1 = (3+4) / 2 = 3.5
Q2 = (6+6) / 2 = 6
Q3 = (8+9) / 2 = 8.5

Calculating the Interquartile Range

The IQR for Dataset B is = 5
IQR = Q3 - Q1
= 8.5 - 3.5
= 5

The variance and the standard deviation are measures of the spread of the data around the mean. They summarise how close each observed data value is to the mean value.

In datasets with a small spread all values are very close to the mean, resulting in a small variance and standard deviation. Where a dataset is more dispersed, values are spread further away from the mean, leading to a larger variance and standard deviation.

The smaller the variance and standard deviation, the more the mean value is indicative of the whole dataset. Therefore, if all values of a dataset are the same, the standard deviation and variance are zero.

The standard deviation of a normal distribution enables us to calculate confidence intervals. In a normal distribution, about 68% of the values are within one standard deviation either side of the mean and about 95% of the scores are within two standard deviations of the mean.

The population Variance σ2 (pronounced sigma squared) of a discrete set of numbers is expressed by the following formula:

where:
Xi represents the ith unit, starting from the first observation to the last
μ represents the population mean
N represents the number of units in the population

The Variance of a sample s2 (pronounced s squared) is expressed by a slightly different formula:

where:
xi represents the ith unit, starting from the first observation to the last
x̅ represents the sample mean
n represents the number of units in the sample

The standard deviation is the square root of the variance. The standard deviation for a population is represented by σ, and the standard deviation for a sample is represented by s.

Which symbol represents the level of significance?

Is an indicator of the precision of an estimate of a statistic?

What is the relationship between clinical significance and statistical significance?

Which of the following is not indicative of qualitative research?