## Understanding the Central Tendencies

Objectives

1. Be able to calculate key values (see below)
2. Be able to use key values within a real world context

Keywords:  Mean, Mode, Median, Maximum, Minimum, Range, Average Range, Outlier

We use Mean, Median and Mode to give us an indication of the middle value for a set of numbers, hence the term central tendencies.

Mean Central Tendency

The mean represents a continuous number and is calculated by add all scores are together then divided the total by the number of scores.  For example:

8 + 5 + 20 + 5 + 9 = 47 ÷ 5 = Mean of 9.4

Examples of continuous data measures:

• Distance
• millimetres
• centimetres
• metres
• Time
• seconds
• minutes
• hours
• Temperature
• degree Celsius
• fahrenheit

Median Central Tendency

The median represents an ordered number (ordinal data, scale response) and is calculated by ranking  or ordering all scores (small to high) then find the middle number.  For example:

5, 5, 8, 9, 20 = Median of 8

Examples of ordinal data measures:

1. likert scale
2. star rating

Mode Central Tendency

The mode represents a category value (categorical data, yes or no) and is calculated by grouping the scores by their value.  The group with the most scores is the mode value.  For example:

[5, 5] [8] [9] [20] = Mode of 5

Examples of categorical data measures:

1. male or female
2. yes or no
3. most prefered item (manufacture, clothing, cars etc…)

Maximum and Minimum Scores

Maximum score is the highest score within your dataset

5, 5, 8, 9, 20  = Max Score of 20

Minimum score is the lowest score within your dataset

5, 5, 8, 9, 20  = Min Score of 5

Range

The range is calculated by subtracting the minimum score from the maximum score.  For example

[5, 5, 8, 9, 20]  20 – 5 = Range of 15

Outlier

An outlier is an isolated score which is very different from the rest.  The outlier is often identified in a box plot graph. For example:

[5, 5, 8, 9, 20]  = Outlier is 20

The distance between the maximum score (20) and the nearest neighbour (9) is 11 compared to the distance between the minimum score (5) and 9 is 4.

Comparing The Central Tendencies (gaming station cabinet)

The findings of this study indicates that the maximum height of a gaming station is 20cm and the minimum is 5cm giving a range of 15cm.  Suggesting that the cabinet space for the gaming station will need to be at least 15cm in height to accommodate most gaming stations.  However, this would consume 20% of the cabinet.

The most commonly occurring height of a gaming station was 5cm which is 4.4cm smaller then the mean value (9.4cm) and 3cm smaller then the median value (8cm). Suggesting that the gaming cabinet space could be reduced to 10cm.  Thus reducing the required space to 13%.

A closer inspection of the data indicates that most (80%, n4) of the gaming stations have a height less than 10cm compared to 20% (n1) of the gaming stations which have a height greater than 10cm.  This study will classify this score of 20cm as an outlier and the central tendencies will be recalculate.  This will tighten the cluster around the mean, giving the results a greater validity or strength.

The new maximum gaming station height is 9cm with an unchanged minimum of 5cm giving a range of 4cm.  Suggesting that the maximum cabinet space for a gaming station is 10cm or 13% of the cabinet.

The most commonly occurring gaming station height is 5cm which is 2.4cm smaller then the mean (7.4cm) and 3cm less them the median (8cm) giving a average range of 3cm.  Suggesting that a cabinet space which is equal to 10.4cm (mean 7.4cm + average range 3cm) which is 1.4cm greater then the maximum height (9cm) will accommodate all gaming stations.

The median (8cm) support the mean (7.4cm) central tendency in suggesting that the cabinet space should be set at the top end of the range as most gaming stations appear to fall there.

Where Next

Distribution and Deviation