Measures of Central Tendency Statistics For Business

MATH0201

“x bar”

The arithmetic mean is the best known type of average and is widely understood. It is used for further statistical analysis.

Advantages Disadvantages

• widely understood •It is easy to calculate •Represents whole set of data •Suited to further statistical analysis

•Value may not correspond to an actual value •Distorted by extreme values

Eg. Mean : ungrouped data

Find arithmetic mean of data in a frequency distribution

x = 185 / 20 = 9.25 = ∑fx / ∑f

Find arithmetic mean of grouped data

• Find mid-point of each class interval.

Arithmetic mean x = ∑ fx / ∑f = 190 / 20 = 9.5 units

The median is the value of the middle item of a distribution once all of the items have been arranged in order of magnitude.

• The median of the following nine values:

8 6 9 12 15 6 3 20 11

• The median of the following ten values:

8 6 9 12 15 6 3 20 11 7

The median of ungrouped frequency distribution

remember to arrange the items in order of value first

The median of a grouped frequency distribution

can be estimated from an ogive

The mode is an average which is the most frequently occurring value.

Advantages Disadvantages

• easy to find •Uninfluenced by a few extreme values •Can be used for non-numerical data

•Does not take every value into account •Some instability in the measure

The mode of a grouped frequency distribution can be estimated from a histogram Modal class of a histogram is always the class with the tallest bar.

Graphical method: using formula

Let L be the lower class boundary of the model class U be the upper class boundary of the modal class Hm be the height of the tallest histogram block Hm- be the height of the block before the tallest one Hm+be the height of the block following the tallest one

Then the mode is given by the formula: Mode = L + Hm - Hm- X (U – L) (Hm - Hm-) + (Hm - Hm+)

Eg.

Invoice value (US$) Frequency, f Frequency density

2 and under 6 1 0.250 6 and under 10 6 1.500 10 and under 14 6 1.500 14 and under 18 10 2.500 18 and under 22 8 2.000 22 and under 30 9 1.125