Measures of
Central Tendency
Objectives to learn:
Objectives to learn:
Measures of Central Tendency
• Arithmetic Mean (Average)
• Median
• Mode
Numerical Data Properties Numerical Data Properties
Central Tendency (Location)
Variation (Dispersion)
Shape
Measures of Central Tendency
Numerical Data Properties & Measures Numerical Data Properties & Measures
Measures of Central Tendency
Numerical Data Properties
Mean Mean
Median Median ModeMode
Central Tendenc
y RangeRange
Interquartile Range Interquartile Range
Variance Variance
Standard Deviation Standard Deviation
Coeff. of Variation Coeff. of Variation
Variation Shap
e
Introduction
❑ Although frequency distributions serve useful purposes, there are many situations that require other types of summarization of data.
❑ What we need in many instances is the ability to summarize the data by means of a single figure called descriptive measure.
❑ Descriptive measures may be computed from the data of a sample or the data of a population.
❑ To distinguish, one from the other, we define them as follows.
Measures of Central Tendency
Several types of descriptive measures can be completed from a set of data
❑ In each of the measures of central tendency, we have a single value that is considered to be typical of a set of data as a value.
❑ In other words a measure of central tendency conveys’ a single information regarding a set of data.
Measures of Central Tendency
❑ By central tendency we mean that the values of the variable in question will cluster around the centre of the series.
❑ The variable concerned must be either a discrete or a continuous one.
❑ Measures of central tendency cannot be found out for qualitative data or variables. For those proportions or percentages can be calculated to analyze data.
Measures of Central Tendency
Three common measures of central tendency are:
i. The arithmetic mean, ii. The median and
iii. The mode.
Measures of Central Tendency
Characteristics of an Ideal Measure of Central Tendency
According to Professor G.U. Yule, a good Average must have the following characteristics:
1. It should be rigidly defined so that different persons may not interpret it differently.
2. It should be easy to understand and easy to calculate.
3. It should be based on all the observations of the data.
4. It should be easily subjected to further mathematical calculations.
5. It should be least affected by the fluctuations of the sampling.
6. It should not be unduly affected by the extreme values.
7. It should be easy to interpret.
Measures of Central Tendency
Mean
⚫ Mean (arithmetic mean) of data values
◦ Sample mean
◦ Population mean
Sample Size
Population Size
Mean
⚫ The most common measure of central tendency
⚫ Affected by extreme values (outliers)
(continued)
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 7 8 9 10 12 14
Mean = 5 Mean = 6
Mean for group data
Age Frequency, f Midpoint
(Age), m f x m
16-20 10 18 10 x 18 =
180
21-25 18 23 18 x 23 =
414
26-30 12 28 12 x 28 =
336
31-35 8 33 8 x 33 = 264
36 -40 2 38 2 x 38 = 76
Formula for group data
Median
⚫Robust measure of central tendency
⚫Not affected by extreme values
⚫In an ordered array, the median is the
“middle” number
◦ If n or N is odd, the median is the middle number
◦ If n or N is even, the median is the average of the two middle numbers
0 1 2 3 4 5 6 7 8 9 10 0 1 2 3 4 5 6 8 10 12 14
Median = 5 Median = 5
Median (when n is odd number)
Median (when n is even number)
Mode
⚫ A measure of central tendency
⚫ Value that occurs most often
⚫ Not affected by extreme values
⚫ Used for either numerical or categorical data
⚫ There may be no mode
⚫ There may be several modes
(multimodal)
0 1 2 3 4 5 6 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14Mode = 9 No Mode
Mode (multimodal)
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