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Discuss the three common measures of location: mean, median, and mode.  What are the advantages and disadvantages of these measures of location?  How does the mean, median, and mode location differ between a symmetric to skewed distribution?

Also, please comment to 2 posts:

POST 1 – The mean is known as the arithmetic average and is denoted by Xbar. It is the most commonly used measure of central tendency and reports the average score of distribution, and is calculated by dividing the sum of the scores by the number of scores, N. It is measured for the interval-ratio level of measurement.

Xbar=summation Xi/N, where Summation Xi is the sum of the scores.

Characteristics:

1. Mean balances the scores, that is if one subtracts the mean from each score, and sums the differences, it will result in 0.

2. The mean minimizes variation of the scores. Since meaning is the point around which variation of the scores is minimized, if the differences between scores and the mean are squared and added, then the resultant sum will be less than the sum of the squared differences between the scores and any other point in the distribution.

3. Mean is misleading if the distribution has outliers.

The Median is the exact center of the distribution of scores. To find the median, array the scores in ascending order. If N is odd, the median will be a score of the middle case and if N is even, the median is halfway between the scores of two middle cases. Median I measured at the ordinal level of measurement.

Mode is the score in a distribution that occurs most frequently. Mode is the only measure of central tendency for a nominal level of measurement.

The mean and the median has the same value for a symmetrical distribution. For a positively skewed distribution, the mean is greater than the median. For a negatively skewed distribution, the mean is less than the median.

A dataset that comprises annual person-hours of time lost due to traffic congestion over the group of cities in the US has data ranging from annual person-hours lost from 10hrs to 50 hrs. This data is skewed in nature, with the possibility of one or more outliers, and therefore only measure of central tendency is not sufficient.

POST 2 – The three common measures of location are mean, median, and mode. Mean is the sum of the data points divided by the number of data points. Calculating the mean is similar to calculating the averages of numbers. Median is the value of the point that has half of the data smaller than that point and half the number larger than that point. Mode is the value of the random sample that occurs with the greatest frequency.

Mean    Takes account of all values to calculate the average.    Very minute or very large values can affect the mean.

Median    The median is not affected by very large or very small values.    Since the median is an average of position, therefore arranging the data in ascending or descending order of magnitude is time-consuming in the case of a large number of observations.

Mode    The only averages that can be used if the data set is not in numbers.    There can be more than one mode, and there can also be no mode which means the mode is not always representative of the data.

The locatiodiffer between a symmetric to skewed distribution becasue:

If the distribution of data is skewed to the left, the mean is less than the median, which is often less than the mode.

(median < median < mode)

If the distribution of data is skewed to the right, the mode is often less than the median, which is less than the mean.

(mean > median > mode)

If the distribution of data is symmetric, the mode = the median = the mean.

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