Measures of Central Tendency

Definition Measures of Central Tendency (Mean, Median, Mode)

Central Tendency Refers to a characteristic where the frequency of a variable tends to cluster around the ‘center’

Measures of Central Tendency Arithmetic Mean Median Mode

Arithmetic Mean Data (units produced by workers) 10, 20, 30 Mean = Ungrouped data (1) =20

Arithmetic Mean Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Ungrouped data (2) Units (x)Worker(f) Total fx

Arithmetic Mean Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Grouped data UnitsWorker(f) 10 – – – – 503 Total Midpoint(m) fm

Arithmetic Mean Ungrouped data Grouped data

Features of Arithmetic Mean Commonly used Easily understood Greatly affected by extreme values

Median 1. Array 2. Median position 3. median

Median Data (units produced by workers) 20, 10, 30 (odd) Ungrouped data (1) ① Array 10, 20, 30 ② Median position ③ Median 20

Median Data (units produced by workers) 20, 10, 40, 30 (even) Ungrouped data (1) ① Array 10, 20, 30, 40 ② Median position ② Median

Median Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Median position= Ungrouped data (2) Units (x)Worker(f) Total12 25 unitsMedian= c.f

Median Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Median = Grouped data (2) UnitsWorker(f) 10 – – – – 503 Total12 Median position = Median Class = c.f

Median Ungrouped data Grouped data

Features of Median Not affected by extreme values When data is skewed, the median is often a better indicator of “average” than the mean. Time consuming Unfamiliar to most people

Mode Data (units produced by workers) 10, 20, 20, 30 Mode = Ungrouped data (1) 20

Mode Data (units produced by workers) 10, 20, 20, 25, 25, 25, 25, 30, 30, 50, 50, 50 Ungrouped data (2) Units (x)Worker(f) Total12 Mode = 25

Mode Data (units produced by workers) 12, 24, 24, 25, 25, 25, 25, 32, 32, 45, 45, 45 Mode = Grouped data (2) UnitsWorker(f) 10 – – – – 503 Total12 The highest frequency: Modal group= units

Mode Ungrouped data Grouped data Data with the highest frequency

Features of Mode Not affected by extreme values May be more than one mode, or no mode May not give a good indication of central values

Skewness of Data Distribution  Normal Mode = mean =median

Skewness of Data Distribution  Positively skewed Mode < median< mean

Skewness of Data Distribution  Negatively skewed Mean < median< mode

Arithmetic Mean ungrouped data grouped data

Median ungrouped data grouped data

Mode ungrouped data grouped data Data with the highest frequency

Measures of Dispersion

Definition Measures of Dispersion(Range, Quartile Deviation, Mean Deviation, Standard Deviation, Variance, Coefficient of Variation)

Dispersion It describes the level of variation and also indicates the level of consistency in the distribution.

Measures of Dispersion Range Quartile Deviation Mean Deviation Standard Deviation Variance Coefficient of Variation

Range It measures the difference between the highest and the lowest piece of data. Data1: Data2: 10, 20, 30 0, 20, 40 Range1 = x max – x min = = 20 Range2 = x max – x min = = 40

Feature It is easy to calculate and easy to understand. It is distorted by extreme values.

Quartile Deviation 1. Array 2. Quartile position 3. Quartile Value 4. IQR,QD

Quartile Deviation It excludes the first and last quarters of information and in doing so concentrates on the main core of data, ignoring extreme values Q1Q2Q3 Interquartile Range = Q 3 - Q1 Quartile Deviation =

Quartile Deviation (ungrouped) Q 1 position= Q 3 position= Q 1 value= Q 3 value=

Grouped data

Amount Spent ($)Number of Staff Total13 c. f

Amount Spent ($)Number of Staff Total13 c. f

Amount Spent ($)Number of Staff Total13 c. f

Amount Spent ($)Number of Staff Total13 c. f

Amount Spent ($)Number of Staff Total13 c. f.

Feature Not effected by extreme values. Not widely used or understood.

Quartile Deviation Q 1 = Q 3 = Ungrouped: I.Q.R= Q 3 value- Q 1 value Quartile Deviation =

Quartile Deviation Q 1 = Q 3 = Grouped: I.Q.R= Q 3 value- Q 1 value Quartile Deviation =

Mean Deviation The absolute distance of each score away from the mean.

Mean Deviation Ungrouped data

Mean Deviation Ungrouped data Team 1: Team 2:

Mean Deviation Ungrouped data Team 1: Team 2:

Mean Deviation Ungrouped data Team 1: Team 2:

Mean Deviation Ungrouped data Team 1: Team 2: M.D. 1 = 2 M.D. 2 =6.4

Mean Deviation Grouped data

UnitsMidpoint(m)Worker(f)fmf|m – | Total ,

Mean Deviation Grouped data Ungrouped data

Standard Deviation/Variance Ungrouped data

Standard Deviation/Variance Ungrouped data Team 1: Team 2:

Standard Deviation/Variance Ungrouped data Team 1: Team 2:

Standard Deviation/Variance Ungrouped data Team 1: Team 2:

Standard Deviation/Variance Ungrouped data Team 1: Team 2:

Standard Deviation (Variance) Grouped data

Units (x) Worker (f) Total Midpoint (m) fm , , ,200 4,450

UnitsMidpoint (m) Worker (f) fmf(m – ) , ,815 Total-502,2004,450

Standard Deviation/Variance Ungrouped data

Standard Deviation (Variance) Grouped data

Coefficient of Variation

Coefficient of Variation (100 Students) Height: Weight: Height C.V.: Weight C.V.: Weight is more variant than Height.

Population & sample