Statistics Fundamentals: Measures of Central Tendency

Measures of Central Tendencies specifies where data is centered (mean, median, and mode).
 
Measures of Location includes (i) measures of central tendency and (ii) other measures that illustrate the location or distribution of data.
 
Arithmetic Mean is the sum of the observations divided by the number of observation
 

• Population Mean (µ) – arithmetic mean value of a population; Formula: µ = (∑ of Xi) / N; Where (i) Xi is the specific value observation and (ii) N is the number of total observations
• Sample Mean – arithmetic mean value of a sample (X bar); Formula: Xbar = (∑ of Xi) / n-1; Where (i) Xi is the specific value observation and (ii) n is the number of total observations
• Data often examined is (i) cross-sectional and (ii) time series data.

 
Properties consists of:

• Its advantages over other measures of central tendency being (a) it uses all the information about the size and magnitude of the observations and (ii) it is easy to work with mathematically;
• Its drawbacks being that it is sensitive to extreme values;
• The sum of the deviations around the mean equals zero, where deviations from the mean helps indicate risk.

 
Median is the value of the middle item of a set of items that has been sorted into either (i) ascending or (ii) descending order
 

• Formula: (I) For odd number of items Median = (n+1)/2, where n = number of observations reading from left to right; (ii) For even numbers Median = the mean of the two corresponding values reading from left to right of (a) n/2 and (b) (n+2)/2.

 
Properties consists of:

• Its advantage being that the median is not sensitive to extreme values;
• Its drawbacks that it does not use all information about (a) the size and (b) magnitude of the observations

Mode is the most frequently occurring value in a distribution. 
 
Properties consists of:

• The modal interval;
• Frequently occurring values such as (a) unimodal (one frequent value), (b) bimodal (two frequent values), and etc.;
• Modal interval being the interval of a data set with the highest frequency

 
Weighted mean can be used for both historical and forward-looking data

• Expected value is the weighted mean of forward looking data (use probability methods to determine weights). Formula is denoted as Xbar = Sum of Xi*wi; where (i) Xi is the specific value observation and (ii) wi is the weight of Xi/ N

 
Geometric mean is most frequently used to average rates of change or to compute the growth rate

• Geometric Mean Formula: G = (X1 * X2 * … Xi)1/n

 
Harmonic mean is the value obtained by summing the reciprocals of the observations

• Special type of weighted mean in which observations weights are inversely proportional to magnitude
• This is used for cost averaging, which involves periodic investment of a fixed amount of money
• Formula: n / ∑ of (1/Xi); where (i) n is the number of observations and (ii) Xi is the specific value observation