**CENTRAL TENDENCY**

**INTRODUCTION**

Apart from the mean, median and mode are the two commonly used measures of central tendency. The median is sometimes referred to as a measure of location as it tells us where the data are. This article describes about median, mode, and also the guidelines for selecting the appropriate measure of central tendency.

**MEDIAN:**

Median is the value which occupies the middle position when all the observations are arranged in an ascending/descending order. It divides the frequency distribution exactly into two halves. Fifty percent of observations in a distribution have scores at or below the median. Hence median is the 50th percentile. j Median is also known as â€˜positional average.

It is easy to calculate the median. If the number of observations are odd, then (n + 1)12^{th} observation (in the ordered set) is the median. When the total number of observations are even, it is given by the mean of n/2th and (n/2 + l) the observation.

**IMPORTANCE/ROLES OF CENTRAL TENDENCY**

Central Tendency is y useful in psychology. It lets us know what is normal or â€˜averageâ€™ for a set of data. It also condenses the data set down to one representative value, which is useful when you are – amounts of data. Could you imagine how difficult it would be to describe the location of a 1,000 item data set if you had to consider every number individual?

Central tendency also allows you to compare one data set to another. For example, letâ€™s say you have a sample of girls and a sample of boys and you are interested in comparing their heights. By calculating the average height for ea ch sample, you could easily draw comparisons between the girls and boys.

Central tendency is also useful when you want to compare one piece of data to the entire data set. Letâ€™s say you received a 60% on your last psychology quiz, which is usually in the D range. You go around and talk to your classmates and find out that the average score on the quiz was 43%. In this instance, your score was significantly higher than those of your classmates. Since your teacher grades on a curve, your 60% becomes an A. Had you not known about the measures of central tendency, you probably would have been really upset by your grade and assume that you bombed the test.

**Three Measures of Central Tendency**

Letâ€™s talk more about the different measures of central tendency. You are probably already familiar with the mean, or average. The mean is calculated in two steps:

- Add the data together to find the sum
- Take the sum of the data and divide it by the total number of data

Now letâ€™s see how this is done using the height example from earlier. Letâ€™s say you have a sample of 10 girls and 9 boys.

The heights in inches are 60, 72, 61, 66, 63, 66, 59, 64, 71, 68. Here are the steps to calculate the mean height for the girls:

First, you add the data together:60+72+61 +66+63+66+59+64+71 +68650.Then, you take the sum of the data (650) and divide it by the total number of data (10 girls): 650 / 10= 65. The average height for the girls in the sample is 65 inches. If you look at the data, you can see that 65 is a good representation of the data set because 65 lands right around the middle of the data set.

The mean is the preferred measure of central tendency because it considers all of the values in the data set. However, the mean is not without limitations. In order to calculate the mean, data must be numerical. You cannot use the mean when you are working with nominal data, which is data on characteristics like gender, appearance, and race. For example, there is no way that you can calculate the mean of the girlsâ€™ eye colors. The mean is also very sensitive to outliers, which are numbers that are much higher or much lower than the rest of the data set and thus, it should not be used when outliers are present.

To illustrate this point, letâ€™s look at what happens to the mean when we change 68 to 680. Again, we add the data together 6o+72+61 +66+63+66+59+64+71 +680=1262. Then we take the sum of the data (1262) and divide it by the total number of data (10 girls): 1262 / 10 = 126.2. The mean height (in inches) for the sample of girls is now 126.2. This number is not a good estimate of the central height for the girls. This number is almost twice as high as the height of n of the girls

However, we can still use other measures of central tendency even when there are outliers. In the scenario above, where a girl who is 680 inches is an outlier, we would use the median. But first, letâ€™s explore how to find a median.

The median is the value that cuts the data set in half If you have an odd number of data, then itâ€™s the value thatâ€™s right in the middle. Letâ€™s practice the boysâ€™ heights since there are 9 boys. There are two steps to finding the median in a sample with an odd number of data:

**Advantages**

- It is easy to compute and comprehend.
- It is not distorted by outliers/skewed data.
- It can be determined for ratio, interval, and ordinal scale.

**Disadvantages**

- It does not take into account the precise value of each observation and hence does not use all information available in the data.
- Unlike mean, median is not amenable to further mathematical calculation and hence is not used in many statistical tests.
- If we pool the observations of two groups, median of the pooled group cannot be expressed in terms of the individual medians of the pooled groups.

**MODE**

Mode is defined as the value that occurs most frequently in the data. Some data sets do not have a mode because each value occurs only once. On the other hand, some data sets can have more than one mode. This happens when the data set has two or more values of equal frequency which is greater than that of any other value. Mode is rarely used as a summary statistic except to describe a bimodal distribution. In a bimodal distribution, the taller peak is called the major mode and the shorter one is the minor mode.

**Advantages**

- It is the only measure of central tendency that can be used for data measured in a nominal scale.
- It can be calculated easily.

**Disadvantages**

- It is not used in statistical analysis as it is not algebraically defined and the fluctuation in the frequency of observation is more when the sample size is small.

**SELECTING THE APPROPRIATE MEASURE**

Mean is generally considered the best measure of central tendency and the most frequently used one. However, there are some situations where the other measures of central tendency are preferred.

Median is preferred to mean when

- There are few extreme scores in the distribution.
- Some scores have undetermined values.
- There is an open ended distribution.
- Data are measured in an ordinal scale.
- Mode is the preferred measure when data are measured in a nominal scale. Geometric mean is the preferred measure of central tendency when data are measured in a logarithmic scale.

**THE ROLE AND SIGNIFICANCE OF MEASURE OF CENTRAL TENDENCY IN DECISION-MAKING**

Measure of central tendency are very useful in statistics, their importance is because of the following reasons.

- To find representative value: Measure of Central Tendency or average gives us one value for the distribution and this value represents the 3entire distribution. In this way averages convert a group of figures into one value.
- To condense data: Collected and classified figure are vast. To condense these figures we use average. Average converts the whole set of figures into just one figure and thus helps in condensation.
- To make Comparisms: To make comparisms of two or more than two distributions, we have to find the representative values of these distributions. These representative values are found with the help of measures of central tendency.
- Helpful in further statistical analysis: Many techniques of statistical analysis like measures of Dispersion, measure of skewness, measure of correlation, and index numbers are based on measure of Central that is why measure of central tendency measure of the first order.

In Â conclusion, different measures of central tendency should be considered by individual cases. There is no definite answer of which to use, but researchers should always choose the one which fits the data the best. If choosing a unsuitable measures, there may be a great variance with the actual result and hence failed to reflect the true finding of the research.

Importance of Measurement Systems Analysis

Measurement Systems analysis is an integral part of the Six Sigma project, No matter what project is being conducted under the Six Sigma methodology, this part of the process can never he omitted out and the successful results still he obtained. Bclow is an pjanation a bout what makes Measurement Systems Analysis such an integral part of the Six Sigma process.

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