Concept of Central Tendency Measures: Meaning, Functions, Characteristics & Types
MEANING OF MEASURES OF CENTRAL TENDENCY
A measure of central tendency describes a summary measure That tries to spell out an entire set of information using one value that reflects the center or center of its supply.
Chaplin (1975) defines central tendency as the representative value of the distribution of scores.
English & English (1958) define measure of central tendency as a statistic calculated from a set of distinct and independent observations and measurements of certain items or entity and intend to typify those observations.
For example, when we talk about the achievement scores of the students of a class, we find some students with very high or very low score. However, the score of the most students live somewhere between the highest and the lowest scores of the whole class. Here we see a score around which the data converge around and this will be used as a measure of central tendency.
FUNCTIONS OF MEASURES OF CENTRAL TENDENCY
Measure of central tendency provides a figure that describes the whole data. It makes it easy for the researcher and the reader to comprehend the data.
It helps in minimizing the large data into a single value. For example, it may not be easy to know a family’s need for electricity, but if we know the average use, the government will plan for generation and procurement of electricity accordingly.
With the help of a sample, it provides us the idea about the mean of the whole population.
It helps in decision making. For example, a telecom company wants to operate in a city and before that it wants to know about average number of calls by a person. Measures of central tendency will be helpful in getting this figure.
Measure of central tendency is used to understand mental functioning. For example, an investigator can use mean score to conclude whether eight-year-old girls are better in linguistic ability in comparison to boys of the same age.
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Definitions of Mean, Mode and Median
Statistical distribution states how a group of data is distributed in a population. For example, if you want to know about the number of children, adolescents and adults in Indian population, we can get the information through the data in a statistical distribution in terms of actual numbers or in terms of percentage. Statistical distribution describes the properties of the distribution in terms of mean, median, mode and range.
There are two types of statistical distributions:
- The discrete random variable distribution, and
- The continuous random variable distribution.
The discrete random variable distribution means variables are usually counts. If a random variable can take only a finite number of distinct values, then it must be discrete. Examples of discrete random variables include the number of children in a family, the Friday night attendance at a cinema, the number of patients in a doctor’s surgery, the number of defective light bulbs in a box of ten.
In continuous random variable distribution, values are recorded within an interval or span. For example, scores of five students are 63, 72, 76, 84, and 39. These are discrete scores. If the same scores are given as 35-55, 55-75 and 75-95.
This is called continuous random distribution. In the continuous distribution, the value will be within an unbroken interval or span. This is also called a probability density function, generally used in the forecasts of weather.
Mean is the average of all values given in both discrete and continuous distribution. It is calculated differently in both discrete distribution and the continuous distribution. In the discrete data, all the scores are added and divided by the total number. In the continuous distribution, there are different methods to calculate the mean.
The Median is the middle value in a series of data. In the discrete data, when the totals of the list are odd, the median is the middle entry in the list after sorting the list into increasing order.
When the totals of the list are even, the median is equal to the sum of the two middle (after sorting the list into increasing order) numbers divided by two. In the continuous distribution, some different formula is applied.
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The mode in a list of numbers refers to the list of numbers that occur most frequently. For example, in the following data —7, 2, 2, 43, 11, 11, 44, 18, 18, 18, 27, 39, 6 -18 occurs the most at 3 times. There can be more than one mode for a distribution with a discrete random variable.
A distribution with two modes is called bimodal and a distribution with three modes is called trimodal. The mode of a distribution with a continuous random variable is calculated differently.
The range of a set of data is the difference between the largest and smallest values. However, in descriptive statistics, this concept of range has a more complex meaning.
It is the same in discrete random variable series and continuous random variable series.
CHARACTERISTICS OF GOOD MEASURES OF CENTRAL TENDENCY
Rightly and rigidly defined: It implies that the definition of the measure should be so clear and same to everyone. It should be interpreted in the same manner.
Simple to calculate: It should lead to one interpretation whoever may be calculating it.
Easy to understand: It should be simple to calculate. Too much complexity and high calculations do not make the measure a good one.
Based on all the observations: It means whatever may be the measure of central tendency, it must be easily understood what it conveys.
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Least affected by fluctuation in sampling: It should be based on all observations means it should take into all the scores. For example, if five students have scored 30, 45, 20, 64, 75 out of 100 marks in a test. All these marks should be considered as it is in the case of Mean. If any mark is left out say the extreme that is 20 and 75, then we cannot calculate the correct measure.
A sample is a smaller group of members of a population selected to represent the population. The most commonly used sample is a simple random sample. It requires that every possible sample of the selected size has an equal chance of being used. For example, we take only 50 in a class of 200 students that is 1/4th of the total class. They represent the whole class.
If we take the sample randomly then this sample will be called random sample. Sample not correctly selected or defective in some way may show fluctuations. For example, if we select only the best student or only the worst, many students will be totally left out who are good or bad? These would affect the measure of central tendency.
If we take the two samples randomly from the same universe or population the value of average for both of them should be near each other. Fluctuation of sampling happens if there difference in the averages of two samples drawn from the same population.
TYPES OF MEASURES OF CENTRAL TENDENCY
There are different measures of central tendency. The three most commonly used are: the mean, median and mode.
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Mean usually refers to average. There are different types of mean such as arithmetic mean, geometric mean and harmonic mean.
The arithmetic mean is the sum of all the scores in a data divided by the total number of scores. M is use for the Mean in psychology, and also recommended by the American Psychological Association. The Greek letter mue is use to denote the mean of the population. X or M is use to denote the mean of a sample.
Properties of the Mean:
Mean is responsive to the changes in any score. Its value will change with increase or decrease in the value of any score.
It is sensitive to the presence or absence of extreme scores. For example, if the data series is 10, 20, 30, 40, the mean is 25 and if we change one score to 20, 30, 40, 50, the mean will 35. One extreme change also changes the mean values drastically.
Among the measures of central tendency, the mean is the best choice. Since it is the only measure which is based on the total scores. For example: if we want to see the impact of training or a group of students. We will compare the mean scores obtained before and after the training.
The difference will give an idea of the effect of training on the students. If the difference is greater than one may be able to state that the training had a good effect on the students.
For an insurance company, on the basis of life expectancy as a Mean the company knows how much it gets from policy-holders and pays to survivors.
The mean is also the most useful when we have to do the further statistical computations.
The mean goes along with other statistical formulas and procedures; it is based on arithmetic and algebraic manipulation. Mean is also includes implicitly or explicitly when we have to do further calculations.
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Limitations of the Mean:
The mean is too responsive to the extreme scores. It means if any one score is extreme and all other scores are near each other, it may give a wrong idea about the average. For example, 5 students scored: 25, 35, 45, 55, and 90. The score of 90 will affect adversely the Mean. Here the mean is 50. If 90 is remove, the mean will become 40.
Every value given equal importance in the calculation of the mean. However, an extreme value becomes misleading.
The mean is often misleading. For example, the average score obtained by group A in first, second and third term is 45, 55 and 50; another group scored 75, 30, 45. The mean in both the cases is 50. We cannot compare the two groups with the mean.
The median is the numerical value separating the higher half of a data sample, a population, or a probability distribution, from the lower half.
Minimum, King & Bear (2001) defines median as the value that divides the distribution into two halves. Garrett (1981) says the median by definition is the 50% point in the distribution when scores in a continuous series are grouped into frequency distribution.
For example, the score of seven students in ascending order is 4, 6, 7, 8, 10, 12; 15, then the marks obtained by the fourth student – 8 – is the median of the scores of the group.
Properties of the Median:
Unlike the mean, median is less responsive to extreme values in the distribution. For example, the median is 8 for 4, 7, 8, 10, 14 and 2, 7, 8, 10, 39.
Sometimes median is also better than the mean as a representative value for a group of scores. For example the mean of the first scores is 12 (4 + 7 + 9 + 14 + 26 5) and the mean of the second scores is 18 (4+7+9+14+54/ 5), but the median is 9 in both the cases, which is much more representative of most of the scores.
An extreme score like this (54) is called an outlier. Outlier may be much higher or much lower than the other scores. Thus, when outliers are present in a data, the Median should used as the central tendency measure.
Limitations of the Median:
The median is misleading as it does not say how many scores lie below or above it. For example, in 10, 25, 30, 49, 50, the median is 30 but the difference between 25 and 30 is 5 and difference between 30 and 49 is 19. Thus, median can be misleading when the data has a very wide range of scores with minimum at one extreme and maximum extreme.
The median does not represent the complete data since it is not based on each and every item of the distribution.
We leave out most of the scores in the median taking only the midpoint value. Thus it cannot be the representative of the sample.
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The mode is the value that appears most often in a set of data. Like the mean and median, the mode is a way of expressing, in a single number, important information, about a random variable or a population.
The word mode originates from the French word La Mode. The literal meaning of the word mode is frequent and fashionable.
Minimum et al (1997) defines mode as the score that occurs with the greater frequency. Guildford (1965) says the mode is strictly defined as the point on the scale of measurement with maximum frequency in a distribution.
For example, in a factory maximum number of laborers (out of 1000, 700 laborers) earn Rs. 200 per day and those earning more than 200 or less than 200, is less than 7. Thus the mode wage of the factory is Rs 200.
Properties of the Mode:
We can easily get the mode in a series of data. The mode can used for nominal level variables. For example, there are more Hindi speaking people in India than people of any other language. Here, Hindi language is referred to the mode.
No other measure of central tendency is appropriate for distribution of language in India, as one can use mode to describe the most common scores in any distribution.
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Limitations of the Mode:
The mode is not stable. It is responsive to sampling fluctuations. In a data series, there may be more than one mode.