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Measures of Center

St. Thomas University

STA 2023

Dr. Freddy Suarez

03/23/2023

Measures of Center

The central position of a set of numbers is represented by its mean. It is calculated by adding the values of all the individual observations' measurements (Urdan, 2022). As a first step in calculating the mean, a set of numbers is sorted, and the middle number is identified (James et al., 2013). The average, or mean, of a group of numbers, is calculated using arithmetic. In this way, the mean locates the midpoint between all measurements. We take the following set of data for the marks scored by 9 students in a certain test: 12, 16,18,15,16,17,19,16 and 15. We add the marks for each student and divide them by the number of students. The total marks are 150 divided by 9 students which give a mean of 16.667 marks.

One can find the value that roughly divides the data in half using the median. This means that 50% of the observations fall below the median, and 50% fall above it (Urdan, 2022). With a set of data with an odd number of observations, the median represents the midway number (Urdan, 2022). Thus, by sorting the observations according to the measurement, the median locates the most central observation in statistical data. Using the above example, we arrange the data in an ascending order as follows: 12, 15, 15,16,16,16,17,18,19 and take the middle number to be the middle number as the median of the data. In this case 16 is the median of the data.

The center of the data set is represented by the mode, which is based on the frequency distribution of the observations (Urdan, 2022). In order to determine which data value occurs

most frequent, we first count how often each value occurs. As a result, the most common occurrence(s) are the modes and a measure of center (James et al., 2013).In this case the number occurring most is 16 which appeared three times.

The mathematical definition of the midrange is the midpoint between the highest and lowest values found in a set of measurements. It is determined by adding the smallest and largest numbers and dividing by 2, yielding the median. Therefore, the midrange can roughly estimate the observation's centre based on the data's extremes. In this case the lowest and the highest numbers are 12 and 19 respectively. We add them to get a midrange of 15.5 marks.

References

James, G., Witten, D., Hastie, T., & Tibshirani, R. (2013). An introduction to statistical learning (Vol. 112, p. 18). New York: springer. ISBN: 978-1-0716-1418-1

Urdan, T. C. (2022). Statistics in plain English. Taylor & Francis. ISBN 9781138838345

Mean, median, mode, and midrange.

Glenda Garrido Blanco

St. Thomas University

STA-2023-AP1-Applied Statistics

Dr. Freddy Suarez

March 22, 2023

Mean, median, mode, and midrange.

Measures of central tendency are used in statistics to describe the typical or central value of a dataset. The mean, median, mode, and midrange are the most commonly used measures of central tendency. Each of these measures provides unique information about the dataset and can be useful in a variety of contexts. The mean is computed by adding all of the values in the dataset and dividing by the total number of values. It is influenced by outliers and extreme values and may not accurately reflect the dataset's average value. It is a useful measure, however, when the dataset has a normal distribution, which means that the majority of values are clustered around the mean (Philipps,2022).

When the values in a dataset are arranged in order, the median is the value in the middle. It is less sensitive to outliers than the mean because it only considers the middle value and not the values of the other data points. This makes it useful when the dataset contains outliers. The mode is the most frequently occurring value in the dataset. When analyzing categorical or nominal data, such as the frequency of different colors in a dataset, it is useful. However, it is possible that it is not unique or that it does not exist in all datasets. The midpoint is the average of the dataset's highest and lowest values. It is a straightforward measure of central tendency, but it is heavily influenced by outliers and extreme values.

I conducted a study on the heights of 50 adults to demonstrate the differences between these measures. The individuals' heights ranged from 4 feet 10 inches to 6 feet 4 inches. The sample was clustered around this height, as the mean height was 5 feet 7.16 inches. There were a few outliers, such as a person who was 4 feet 10 inches tall, who skewed the mean (Mohan & Su,2022).

The median height was 5 feet 7.5 inches, which was close to the average but less influenced by outliers. The most common height in the dataset was 5 feet 6 inches, which was the mode height. This could be useful information if we were looking for the sample's average height. The median height was 5 feet 7 inches, which was a simple average of the dataset's highest and lowest values. It was, however, heavily influenced by the outlier at 4 feet 10 inches.

In conclusion, the mean, median, mode, and midrange are all central tendency measures that provide different information about a dataset. The type of measure used is determined by the type of data and the research question. Before using a measure to analyze data, it is critical to understand its strengths and limitations.

References.

Mohan, S., & Su, M. K. (2022). Biostatistics and Epidemiology for the Toxicologist: Measures of Central Tendency and Variability—Where Is the “Middle?” and What Is the “Spread?”.  Journal of Medical Toxicology18(3), 235-238.

Philipps, C. (2022). Interpreting expectiles.  Available at SSRN 3881402.

Edited by  Garrido Blanco, Glenda  on Mar 23 at 7:25pm