Which measure of variability indicates the difference between the highest and lowest score?

• Range
• Interquartile range
• Variance
• Standard deviation

Range

The most basic measure of variation is the range, which is the distance from the smallest to the largest value in a distribution.

Range= Largest value – Smallest Value

For the distribution of scores of Quiz 1 and Quiz 2, the range is:

Quiz 1. Range = 9-5= 4

Quiz 2. Range = 10-4= 6

which shows (like the histograms above) that Quiz 2 scores have greater spread than Quiz 1 scores.

However, the range uses only two values in the data set, and one of these values may be an unusually large or small value.

Interquartile range

The interquartile range (IQR) is the range of the middle 50% scores in a distribution:

IQR= 75th percentile  – 25th percentile

It is based on dividing a data set into quartiles. Quartiles are the values that divide scores into quarters. Q1 is the lower quartile and is the middle number between the smallest number and the median of a data set. Q2 is the middle quartile-or median. Q3 is the upper quartile and is the middle value between the median set and the highest value of a data set. The interquartile range formula is the first quartile subtracted from the third quartile

For Quiz 1, Q3 is 8 and Q1 is 6 . These are the scores:

5, 6, 7, 8, 9

If the median is 7, then Q1 is 6 (middle value between median and lowest value) and Q3 is 8 (middle value between median and highest value).

To calculate the IQR:

IQR= 8-6= 2

For Quiz 2, Q3 is 9 and Q1  is 5. These are the scores:

4, 5, 6, 7, 8, 9, 10

The median is 7. To find Q1, we’ll look at the lower half section of the distribution of scores: 4,5,6. Q 1 is the median of this section of the distribution: 5

To find Q2, we’ll look at the upper half section of the distribution of scores: 8, 9,10. Q3 is the median of this section of the distribution: 9.

To calculate the IQR, knowing Q1 and Q3:

IQR= 9-5= 4

Variance

The variance is the average squared difference of the scores from the mean. To compute the variance in a population:

1. Calculate the mean
2. Subtract the mean from each score to compute the deviation from mean score
3. Square each deviation score (multiply each score by itself)
4. Add up the squared deviation score to give the sum
5. Divide the sum by the number of scores

The table below contains students’ scores on a Statistics test. To calculate the variance:

1. The mean is calculated: sum all scores and divide by the number of scores:   140/20= 7
2. The deviation from the mean for each score is calculated. For example, for the first score: 9-7= 2- See column Deviation from the mean
3. Each deviation from the mean score is squared (multiplied by itself). For the first score: 2x2= 4. See column Squared deviation.
4. Finally, the mean of the squared deviations is calculated. The variance is 1.5

This is how the formula to calculate variance in a population looks like:

Where o2 is the variance

µ is the mean of a population

X are the values or scores

N is the number of values or scores

If the variance in a sample is used to estimate the variance in a population, it is important to note that samples are consistently less variable than their populations:

• The sample variability gives a biased estimate of the population variability.

• This bias is in the direction of underestimating the population value.
• In order to adjust this consistent underestimation of the population variance, we divide the sum of the squared deviation by N-1 instead of N.

Formula to calculate variance in a sample is:

Where s2 is the variance of the sample

M is the sample mean

X are the values or scores

N is the number of values or scores in the sample

Standard deviation

The standard deviation is the average amount by which scores differ from the mean. The standard deviation is the square root of the variance, and it is a useful measure of variability when the distribution is normal or approximately normal (see below on the normality of distributions). The proportion of the distribution within a given number of standard deviations (or distance) from the mean can be calculated.

A small standard deviation coefficient indicates a small degree of variability (that is, scores are close together); larger standard deviation coefficients indicate large variability (that is, scores are far apart).

The formula to calculate the standard deviation is

Note that the standard deviation is the square root of the variance.

Example: how to calculate the standard deviation:

In the previous section- Variance- we computed the variance of scores on a Statistics test by calculating the distance from the mean for each score,t hen squaring each deviation from the mean, and finally calculating the mean of the squared deviations.

Since we already know the variance, we can use it to calculate the standard deviation. To do so, take the square root of the variance. The square root of 1.5 is 1.22. The standard deviation is 1.22.

Distributions with the same mean can have different standard deviations. As mentioned before, a small standard deviation coefficient indicates that scores are close together, whilst a large standard deviation coefficient indicates that scores are far apart. In this example, both sets of data have the same mean, but the standard deviation coefficient is different:

In this example, the scores in Set A are 0.82 away from the mean; in Set B, scores are 2.65 away from the mean, even though the mean is the same for both sets. So scores in Set B are more dispersed than scores in Set A.

Activity 1- Mean and standard deviation

Interpret means and standard deviation

1. Which group showed higher levels of aggressive behaviour?

      Males

      Females

      Males in the control group

      Females in the control group

2. Which group- exposed to violent cartoons- presents more variability in their scores?

      Females

      Males

1. Which group showed higher levels of aggressive behaviour?

Males

Females

Males in the control group

Females in the control group

The highest mean is 15.72- males exposed to violent cartoons

2. Which group- exposed to violent cartoons- presents more variability in their scores?

Females

Males

Standard deviation coefficient of the group 'males exposed to violent cartoons' is higher than the sd coefficient of the group 'females exposed to violent cartoons'

Which measure of variability shows the difference between the highest and the lowest score?

What does high and low variability mean?

What is the strongest measure of variability?