Teaching Measures of Central Tendency

This paper provides a descriptive narration of Measures of Central Tendency (the mean, the median, the mode, the weighted mean and the distribution shapes) with solved examples to illustrate these measures. As the paper describes, measures of central tendency is a category of descriptive analysis, which uses a single value to describe the central representation of any dataset and thus a useful tool in analysis. Due to the disparities that happen to be in different data sets, the mean or the average by itself may not provide the needed information about the distribution of the data. Therefore, the different measures of central tendency give adequate information concerning the distribution of any data set thus important to understand them.

Teaching Measures of Central Tendency

Measures of central tendency is one of the two categories of descriptive statistics that uses a single value as a central representation of a data set and it is important in statistical analysis as it represents a large set of data using only one value. From this category of description, several methods apply to represent this central part. Among the measures includes mean, median, mode, weighted mean and description shapes. The methods of analysis are crucial in statistical analysis as they give information of any data set of interest.

First, we examine the mean as measure of central tendency. Being the commonly utilized measure, it takes another name as average and it involves calculation of summing up all values in a selected population and then dividing the total sum by the involved number of observations. Depending on the desired mean, sample mean, or population mean, the resulting formula can differ slightly. All the same, the result is a central representation of a data set. For instance, if a data set constitutes the following 5 observations, 2, 7, 4, 9,and 3, then the mean will be obtained by summing up all observations (2 + 7 + 4 + 9 + 3) to obtain a cumulative sum of 25, then dividing this result with the number of observations (Mean = 25/5 = 5). Therefore, the mean of the five observations is equal to five (Donnelly, 2004, p. 46).

The next measure is the weighted mean. Unlike the normal mean or average, which allocates equal weight to all values of the observation, weighted mean gives the flexibility to allocate more weight on certain values of the observation compared to others. For example, considering the scores of a student in three exams, that constitutes the exam having a 50% weight, practical contributing 30% weight while the homework takes the remaining 20% weight. If this student scores 80, 70 and 65 in exam, practical and homework respectively, then the weighted mean of these scores obtainable. This is possible through summing up the products of exam score and its respective weight, then dividing the result by to total sum of the three weights, (weighted mean = ((50*80)+(70*30)+(65*20))/(50+30+20))=74) (Salkind, 2009, p. 46).

The third measure of central tendency is the mode. Despite it being the last option of consideration by many analysts, it is a mostly utilized measure. The mode represents the most frequent observation in a data set. For example, if total scores of a football tournament in every match were tabulated as 2, 4, 6, 5, 2, 4, and 2, then the mode of these observations is 2 scores because this observation occurs thrice in the distribution.

The median contributes to the fourth measure of central tendency and it represents the value in the distribution that lies in the middle of the observations of interest. To obtain the median, arranging the observations in ascending order establishes the observation that lies in the middle of the arranged data. For example, in the above given observations of football scores, they can arranged in ascending order as 2, 2, 2, 4, 4, 5, 6, then looking at this order, 4 is the median of the observation as it lies mid the arrangement. In the event there happens to be an even distribution, then the median is the average of the two observations lying in the middle of the ascending arrangement (Donnelly, 2004, p. 44).

The fifth and the final measure in this paper are the distribution shapes and these will give a visual representation of central point of a distribution. The distribution shapes includes, normal or symmetrical distributions, skewed distributions constituting positively skewed distribution and negatively skewed distributions, bimodal distribution and uniform distributions. The symmetrical distribution takes the shape of a bell curve when one plots the observations and the mean lies in the center of the bell curve splitting the tails evenly. In addition, the mode is equivalent to the mean of the distribution. The skewed distributions usually involve a data set that has extreme outliers resulting to asymmetrical bell curve and the observation will tend to concentrate at either of the tail ends showing a lopsided distributions. These skewed distributions have the positively skewed distributions where the greatest amount of observations lies on the left side of the plotted distribution while negatively skewed distribution has most of its outliers concentrated in the far right. On its part, bimodal distribution will represent a two-humped distribution while a uniform distribution remains a rare observation because in nature it is difficult to observe same frequencies for different responses (Pathways courses, 2003).

In conclusion, the measures of central tendency give information of the distribution of observations that represents the data effectively and are important measures in analysis. Apart from the commonly used mean, mode, median, weighted mean and distribution shapes gives a different option to obtain information concerning distribution of any data set. However, despite all these methods giving this important information, many analysts prefer the mean or average that the mode. Nevertheless, they are all measures of central tendency and help to obtain information of the central distribution of data.

References

Distributions. (2003, May 11). Wading through the data swamp: Distributions. Pathway courses.

Retrieved from http://pathwayscourses.samhsa.gov/eval201/eval201_supps_pg16.htm#Anchor-Normal-49575

Donnelly, R.A. (2004). The complete idiot’s guide to statistics. Indianapolis, in: Alpha books.

Salkind, N.J. (2009). Statistics for people who (think they) hate statistics, (2nd ed.). London, UK:

SAGE.

This paper provides a descriptive narration of Measures of Central Tendency (the mean, the median, the mode, the weighted mean and the distribution shapes) with solved examples to illustrate these measures. As the paper describes, measures of central tendency is a category of descriptive analysis, which uses a single value to describe the central representation of any dataset and thus a useful tool in analysis. Due to the disparities that happen to be in different data sets, the mean or the average by itself may not provide the needed information about the distribution of the data. Therefore, the different measures of central tendency give adequate information concerning the distribution of any data set thus important to understand them.

Teaching Measures of Central Tendency

Measures of central tendency is one of the two categories of descriptive statistics that uses a single value as a central representation of a data set and it is important in statistical analysis as it represents a large set of data using only one value. From this category of description, several methods apply to represent this central part. Among the measures includes mean, median, mode, weighted mean and description shapes. The methods of analysis are crucial in statistical analysis as they give information of any data set of interest.

First, we examine the mean as measure of central tendency. Being the commonly utilized measure, it takes another name as average and it involves calculation of summing up all values in a selected population and then dividing the total sum by the involved number of observations. Depending on the desired mean, sample mean, or population mean, the resulting formula can differ slightly. All the same, the result is a central representation of a data set. For instance, if a data set constitutes the following 5 observations, 2, 7, 4, 9,and 3, then the mean will be obtained by summing up all observations (2 + 7 + 4 + 9 + 3) to obtain a cumulative sum of 25, then dividing this result with the number of observations (Mean = 25/5 = 5). Therefore, the mean of the five observations is equal to five (Donnelly, 2004, p. 46).

The next measure is the weighted mean. Unlike the normal mean or average, which allocates equal weight to all values of the observation, weighted mean gives the flexibility to allocate more weight on certain values of the observation compared to others. For example, considering the scores of a student in three exams, that constitutes the exam having a 50% weight, practical contributing 30% weight while the homework takes the remaining 20% weight. If this student scores 80, 70 and 65 in exam, practical and homework respectively, then the weighted mean of these scores obtainable. This is possible through summing up the products of exam score and its respective weight, then dividing the result by to total sum of the three weights, (weighted mean = ((50*80)+(70*30)+(65*20))/(50+30+20))=74) (Salkind, 2009, p. 46).

The third measure of central tendency is the mode. Despite it being the last option of consideration by many analysts, it is a mostly utilized measure. The mode represents the most frequent observation in a data set. For example, if total scores of a football tournament in every match were tabulated as 2, 4, 6, 5, 2, 4, and 2, then the mode of these observations is 2 scores because this observation occurs thrice in the distribution.

The median contributes to the fourth measure of central tendency and it represents the value in the distribution that lies in the middle of the observations of interest. To obtain the median, arranging the observations in ascending order establishes the observation that lies in the middle of the arranged data. For example, in the above given observations of football scores, they can arranged in ascending order as 2, 2, 2, 4, 4, 5, 6, then looking at this order, 4 is the median of the observation as it lies mid the arrangement. In the event there happens to be an even distribution, then the median is the average of the two observations lying in the middle of the ascending arrangement (Donnelly, 2004, p. 44).

The fifth and the final measure in this paper are the distribution shapes and these will give a visual representation of central point of a distribution. The distribution shapes includes, normal or symmetrical distributions, skewed distributions constituting positively skewed distribution and negatively skewed distributions, bimodal distribution and uniform distributions. The symmetrical distribution takes the shape of a bell curve when one plots the observations and the mean lies in the center of the bell curve splitting the tails evenly. In addition, the mode is equivalent to the mean of the distribution. The skewed distributions usually involve a data set that has extreme outliers resulting to asymmetrical bell curve and the observation will tend to concentrate at either of the tail ends showing a lopsided distributions. These skewed distributions have the positively skewed distributions where the greatest amount of observations lies on the left side of the plotted distribution while negatively skewed distribution has most of its outliers concentrated in the far right. On its part, bimodal distribution will represent a two-humped distribution while a uniform distribution remains a rare observation because in nature it is difficult to observe same frequencies for different responses (Pathways courses, 2003).

In conclusion, the measures of central tendency give information of the distribution of observations that represents the data effectively and are important measures in analysis. Apart from the commonly used mean, mode, median, weighted mean and distribution shapes gives a different option to obtain information concerning distribution of any data set. However, despite all these methods giving this important information, many analysts prefer the mean or average that the mode. Nevertheless, they are all measures of central tendency and help to obtain information of the central distribution of data.

References

Distributions. (2003, May 11). Wading through the data swamp: Distributions. Pathway courses.

Retrieved from http://pathwayscourses.samhsa.gov/eval201/eval201_supps_pg16.htm#Anchor-Normal-49575

Donnelly, R.A. (2004). The complete idiot’s guide to statistics. Indianapolis, in: Alpha books.

Salkind, N.J. (2009). Statistics for people who (think they) hate statistics, (2nd ed.). London, UK:

SAGE.