Important Questions: Measures of Central Tendency

Q1: What is positional average?
Ans: Positional average is an average determined by the position or rank of a value within an ordered dataset rather than by arithmetic computation of all values. Examples of positional averages are median, quartiles and percentiles. These measures are useful for describing the relative standing of observations, for ordinal data and for distributions that are skewed or contain extreme values.

Q2: What are the two methods that can calculate the simple arithmetic mean in the case of an individual series?
Ans: The two methods that can be used to calculate the simple arithmetic mean for an individual series are:

• Direct method: Compute the mean by adding all individual values and dividing by the total number of observations: mean = (sum of all values) ÷ (number of values). This is straightforward and is used when the list of values is manageable.
• Shortcut method:Choose a convenient assumed mean (A), calculate deviations of each observation from A, sum those deviations and divide by the number of observations, then add the result to A. This reduces arithmetic when numbers are large or inconvenient.

Q3: What is mode?
Ans:Mode is the observation that occurs most frequently in a series. A series may be unimodal (one mode), bimodal (two modes) or multimodal (more than two modes). Mode is particularly useful for categorical data where mean and median cannot be defined. In grouped data the mode is represented by the modal class (the class with highest frequency).

Q4: What are the methods of calculating the simple arithmetic mean?
Ans: The methods of calculating the simple arithmetic mean depend on how data are presented:

• Individual series:Add all individual observations and divide by the total number of observations.
• Discrete series:Multiply each distinct value by its frequency, sum these products and divide by the total frequency.
• Frequency distribution:For grouped data, use class mid-points (class marks); multiply each class mid-point by its class frequency, sum these products and divide by the total frequency to obtain the mean.

Q5: Define the partition value.
Ans: The partition value is a numerical point that divides an ordered series into a specified number of parts. Common partition values include quartiles (divide data into four parts), deciles (ten parts) and percentiles (one hundred parts). Partition values help to study portions of a distribution, compare segments and identify where particular observations lie.

Q6: What are the different kinds of statistical average?
Ans: The main kinds of statistical average are:

• Mathematical averages:These are computed by formulae and include mean, median and mode.
• Positional averages:These are based on the position of observations in an ordered list and include percentiles, quartiles and other rank-based measures.

Q7: Explain quartile.
Ans: A quartile divides a data set ordered from smallest to largest into four equal parts. The three quartiles are:

• Q1: the first quartile or 25th percentile (25% of observations are below Q1);
• Q2: the second quartile or 50th percentile, which is the median;
• Q3: the third quartile or 75th percentile (75% of observations are below Q3).

Quartiles are used to describe the spread of the middle portion of the data; the difference Q3 - Q1 is called the interquartile range (IQR) and measures the central dispersion.

Q8: Define the central tendency.
Ans: Central tendency denotes a statistical value that represents the centre or typical value of a distribution. Common measures are mean, median and mode. These measures summarise a large set of observations by a single representative value and are widely used for comparison, reporting and decision making.

Q9: What are the purposes of average in the statistical method?
Ans: The purposes of using an average in statistics are:

• To provide a concise and representative summary of a dataset.
• To enable easy comparison between different datasets or groups.
• To assist policy makers and managers in making informed decisions.
• To facilitate analysis by reducing complex data to a single, interpretable value.
• To present a central value that represents the entire dataset for reporting and interpretation.

Q10: Define median.
Ans: The median is the middle value in an ordered series of observations. If the number of observations is odd, the median is the central observation. If the number is even, the median is the average of the two central observations. The median divides the series into two halves so that half the values lie below it and half lie above it, and it is less influenced by extreme values than the mean.