MEASURES OF CENTRAL TENDENCY Flashcards
What is a measure of central tendency
A measure of central tendency is a number that locates the approximate center of a distribution of data. The purpose of a measure of central tendency is to locate the “average” or “typical” case in a distribution of cases.
Define the term ‘average’ in statistics
that value of a distribution which is considered as the most representative or typical value for a group.
Give examples of measures of central tendency
The most commonly used measures of central tendency or averages are the mean, mode and median. Other types of averages include the weighted arithmetic mean, trimmed mean, geometric mean and harmonic mean.
What are the two main objectives of averaging
- To get a single value that describes the characteristics of the entire group
- To facilitate comparison between groups.
What are Properties of a Good (Average)Measure of Central tendency
- I t should be easy to understand. It should be readily understood otherwise its use will be limited.
- It should be simple to compute. It is important to note though that ease of computation should not be at the expense of other advantages.
- It should be based on all items. It should depend on each and every item of the data set so that if any item is dropped its value is altered.
- It should not be unduly affected by extreme items. If one or two very small or very large items unduly affect the average then it cannot be typical of the entire data set. Extremes may distort its value and reduce its usefulness
- It should be rigidly defined. An average should be properly defined preferably by an algebraic formula so that it has only one interpretation. Different people computing it from the same figures should get the same answer.
- It should be capable of further algebraic treatment. It can be used for further statistical computations to enhance its usefulness..
- It should have sampling stability. If we pick different samples from a population and compute the average for each of them we should expect to get approximately the same value.
What is mean, sample mean and population mean and how are they calculated
- See notes for formulas
What are the properties of the mean
- It is the point in a distribution of measurements or scores about which the sum of the
deviations are equal to zero.
The mean is therefore characterized as a point of balance. It is the value that balances all scores on either side of it. *See notes - The sum of the square of the deviations of the items from the arithmetic mean is minimum, that is it is less than the sum of squared deviations of items from any other value. This property is of immense use in regression analysis which is topic to be covered later.
- If each item in a series is replaced by the mean, then the sum of these substitutions will be equal to the sum of the individual items.
- Using the arithmetic mean and number of items of two or more related groups we can compute the combined mean
* See notes for further explanations and formulas
What are the merits of the arithmetic mean
- It is the simplest average to understand and easiest to compute
- Its computation is based on all items of the series
- It is rigidly defined. Every one computing the arithmetic mean for the same data set will get the same answer.
- It lends itself to further algebraic treatment.
- It is relatively reliable in the sense that it does not vary too much when repeated samples are taken from one and the same population.
What are the limitations of arithmetic mean
- The mean is very sensitive to extreme values when these are not evenly dispersed on both sides of it. E.g Comparing two data sets 2,3,5,7,8 and 2,3,5,7,33. The mean for the first one is 5, while that for the second one is 10. The large score of 33 in the second group makes the mean of the second group double. When a distribution is markedly skewed the mean provides a misleading measure of central tendency. The mean provides a an appropriate ‘average’ , only when the distribution of a variable is reasonably normal(bell shaped) Income is a commonly studied variable in which the median is preferred over the mean , since the distribution is distinctly skewed in the direction of high incomes.
- The mean cannot be computed in a distribution with open ended classes without making assumptions regarding the size of the class interval of the open ended classes which may lead to substantial errors.
What is the median?
The median is the half way point in a data set. It is the point at which 50% of the values in the data set have a value the size of the median value or smaller and 50% of the values have a value the size of the median value or larger.
How do you calculate median?
*See notes
What are the properties and merits of the median
Properties of the median:
It is insensitive to extreme scores
Merits
- Useful incase of open ended classes
- Not influenced by extreme scores and therefore is preferred to the arithmetic mean in skewed distributions such as income
- It is most appropriate when dealing with qualitative data i.e where ranks are given or there are other types of items that are not counted or measured but are scored.
- The median can be determined graphically
What are the limitations of the median
1.Its computation does not involve all items in the series
2.It is not capable of further algebraic treatment e.g. we cannot find the combined median of
two data sets.
3. Its value is affected more by sampling fluctuations than the value of the arithmetic mean.
How do you calculate mode?
*See notes
What are the different types of means
Distributions that have only one mode are referred to as unimodal. Those with two, three or more mode are referred to as bimodal, trimodal or multimodal respectively.
