Centre or Spread ?

Measures of Central Tendency

Measures of central tendency are tools we use to give us an estimate of what score represents the centre of a data set. We've looked at different measure of central tendency, particularly the:

• mean: the average of the score,
• median: the middle score, and
• mode: the most frequently occurring score.

 

Measures of Distribution

Measures of distribution tell us how far the scores in a data set are spread out.

We've already looked a one measure of spread, the range, which is the difference between the highest and lowest score in a data set.

Now we are going to learn about a new measure of spread called the Mean Absolute Deviation (MAD). The MAD of a set of data is the average distance between the scores and the mean. 

Calculating the MAD of a data set

Let's use an example to help explain this.

Find the mean absolute deviation of {$23,18,31,28,20$23,18,31,28,20}.

1. Find the mean.

$Mean=\frac{23+18+31+28+20}{5}$Mean=23+18+31+28+205​$=$=$24$24

2. Find the difference between each individual score and the mean.

$23-24=-1$23−24=−1

$18-24=-6$18−24=−6

$31-24=7$31−24=7

$28-24=4$28−24=4

$20-24=-4$20−24=−4

3. Take the absolute value of each difference.

$\left|-1\right|=1$|−1|=1

$\left|-6\right|=6$|−6|=6

$\left|7\right|=7$|7|=7

$\left|4\right|=4$|4|=4

$\left|-4\right|=4$|−4|=4

Find the mean of these differences.

$Mean=\frac{1+6+7+4+4}{5}$Mean=1+6+7+4+45​$=$=$4.4$4.4

This means that, on average, scores in this data set are $4.4$4.4 units above or below the mean.

 

Examples 

Question 1

A batsman’s mean number of runs is $57$57 and the MAD is $13$13. In the next match he makes $58$58 runs. If this score is added to the existing scores, which is true of the new mean and MAD?

A) $\text{Mean }<57$Mean <57, with $MAD<13$MAD<13     B) $\text{Mean }>57$Mean >57, with $MAD>13$MAD>13     C) $\text{Mean }<57$Mean <57, with $MAD>13$MAD>13     D) $\text{Mean }>57$Mean >57, with $MAD<13$MAD<13

Think: Is the new score lower or higher than the old mean? Also, are the other scores are, on average, further from the old mean?

Do: D) $Mean>57$Mean>57, with $MAD<13$MAD<13

 

Question 2 (

 

Question 3