Managerial Statistics

Assessment 1: Individual Assignment
Course:
Name:
ID:
Table of Contents TOC \o "1-3" \h \z \u Abstract PAGEREF _Toc87987062 \h 31.Introduction PAGEREF _Toc87987063 \h 42.Histogram, Frequency Polygon, and Cumulative Frequency Polygon PAGEREF _Toc87987064 \h 42.1Histograms PAGEREF _Toc87987065 \h 42.2Frequency Polygons PAGEREF _Toc87987066 \h 52.3Cumulative Frequency Polygons PAGEREF _Toc87987067 \h 63.Arithmetic Mean, Mean Deviation, Standard Deviation, and Skewness PAGEREF _Toc87987068 \h 73.1Arithmetic Mean PAGEREF _Toc87987069 \h 73.2Mean Deviation PAGEREF _Toc87987070 \h 73.3Standard Deviation PAGEREF _Toc87987071 \h 83.4Skewness PAGEREF _Toc87987072 \h 84.Median, Quartiles, and Box Plot PAGEREF _Toc87987073 \h 84.1Median PAGEREF _Toc87987074 \h 84.2Quartile PAGEREF _Toc87987075 \h 94.3Box Plot PAGEREF _Toc87987076 \h 95.Advantages, Disadvantages, and Uses through SPSS PAGEREF _Toc87987077 \h 95.1Histogram, Frequency Polygon, and Cumulative Frequency Polygon PAGEREF _Toc87987078 \h 105.2Mean, Mean Deviation, Standard Deviation and Skewness PAGEREF _Toc87987079 \h 135.3Median, Quartile and Box Plot PAGEREF _Toc87987080 \h 146.Critical Evaluation PAGEREF _Toc87987081 \h 167.Conclusion and Recommendation PAGEREF _Toc87987082 \h 16Bibliography PAGEREF _Toc87987083 \h 17Appendix: Data Set PAGEREF _Toc87987084 \h 19
Abstract
Management Decisions play a key role in ensuring success for any business. However, management decisions are dependent on the usefulness of available information. Quantitative methods enhance the quality of insights from the information, which can, in turn, improve the quality of management decisions. Suppose the manager of an investment house lacks knowledge of quantitative methods, which could potentially be impacting his recommendations to his clients. Therefore, this report is prepared to give him an overview of several statistical measures: histograms, frequency polygon, cumulative frequency polygon, mean, mean deviation, standard deviation, skewness, median, quartiles, and box plots, with the focus on description, differences, and advantages and disadvantages. Also, the use of these measures is demonstrated using SPSS.
Keywords: Management Statistics, Quantitative Methods, SPSS
1 Introduction
For any business, its success is dependent on the quality of management decisions, which in turn is dependent on the insights available with the management (Radha & Balaji, 2013). Radha and Balaji (2013) highlighted that the past performance of the business is useful information. However, this information can be unfiltered at times, and as a result, meaningful inferences cannot be drawn.
The portfolio manager of an investment house is facing a similar situation as he lacks knowledge of various quantitative methods. According to Gupta and Gupta (2013), there is a possibility that his lack of knowledge could be impacting the quality of decisions and investment recommendations being made by the manager.
Thus, this report is being prepared to give an overview of several statistical measures: histograms, frequency polygon, cumulative frequency polygon, mean, mean deviation, standard deviation, skewness, median, quartiles, and box plots.
This report focuses on describing these measures, identifying differences from similar measures, and discussing advantages and disadvantages. Also, the use of these measures in understanding the monthly return by S&P 500 Index is demonstrated using SPSS.
2 Histogram, Frequency Polygon, and Cumulative Frequency Polygon
1 Histograms
Histogram organizes data distribution into user-specified ranges (or bins). Two important characteristics of bins are that they must be adjacent and non-overlapping. While it is usually preferred to keep the bin size the same, it is not a necessary condition. If bins are of equal size, the rectangle's height is to be used in proportion to the frequency in each bin. However, if bins are not of equal size, the rectangle has an area proportional to the frequency in the bin. In this case, instead of the frequency, the height depicts the frequency density. Histograms can also be normalized to show relative frequencies, i.e., proportion to the number of cases that fall into different categories, with the sum of heights being 1 (Levin et al., 2017).
A sample histogram is presented below:
Figure 1: Sample Histogram (Source: Gupta & Gupta, 2013)
The visual representation makes it easy to understand the data. In a business setting, it is widely used to understand different data distributions. An example where it can be used is to arrange the viewers of a TV show by dividing them into different age brackets (bins) (Levin et al., 2017).
Histograms are often confused with bar charts; however, the major difference is that while bar chart is used for categorical variables, the histogram is used for continuous data (Levin et al., 2017).
2 Frequency Polygons
Frequency Polygon is also a visual representation to understand the shape of distribution as it indicates the frequency in different classes in the dataset (Holmes et al., 2017).
A random frequency polygon is shown below to give an idea to the manager:
Figure 2: Sample Frequency Polygon (Source: Gupta & Gupta, 2013)
As shown in the above figure, the frequency polygon is a curve where the x-axis represents the values of the dataset, and the y-axis shows the frequency in each category. Its usage is similar to a histogram. Therefore it can be used as an alternative to a histogram or can be used simultaneously (Holmes et al., 2017).
3 Cumulative Frequency Polygons
A cumulative frequency polygon is a frequency polygon that shows cumulative frequencies from left to right (Holmes et al, 2017), as shown in the figure on the following page.
Figure 3: Sample Cumulative Frequency Polygon (Source: Gupta & Gupta, 2013)
These graphs are used to understand how many numbers lie above or below a particular value in data. This is constructed by calculating the cumulative frequencies (the frequencies of all the preceding variables in the data set) of variables from a frequency table (Holmes et al., 2017).
A cumulative frequency polygon is similar to a histogram. However, while a histogram uses rectangles, a cumulative frequency polygon uses a single point marking where the top right of the rectangle of the histogram should be (Holmes et al., 2017).
3 Arithmetic Mean, Mean Deviation, Standard Deviation, and Skewness
4 Arithmetic Mean
The arithmetic mean is the most widely used and simplest measure of understanding a dataset by calculating the average value (sum of all data values which is divided by the count of data values) (Anderson, 2013).
5 Mean Deviation
Mean deviation refers to the measure of computing the difference between the mean and other values in the data set. It can be calculated for both ungrouped and grouped data. It calculates the average absolute deviation of values from the mean. For a data set of 2,7,5, and 10, the mean is 6 and mean average deviation is ((6-2) +(7-6) +(6-5) + (10-6))/4 = 2.5. While the arithmetic means only show the average value, the mean deviation also shows how the values are dispersed in the data set (Black, 2012).
6 Standard Deviation
Standard deviation is also a measure of the dispersion of the dataset related to the mean. It is calculated as the square root of the variance (calculated by determining the deviation for each value); further are data points from the mean, higher is the standard deviation. While mean deviation ignores the modularity and only considers the absolute value, standard deviation also considers the modularity. Therefore, the standard deviation is a better measure for dispersion in comparison to the mean deviation (Levin et al., 2017).
7 Skewness
Skewness is a measure to understand the probability distribution of random variables around its mean. Consider a business whose 50% of staff is below the age of 25, and 65% of the staff is below the age of 35. So, if this age distribution is plotted, then it can be observed that there is a hump on the left side of the distribution, and the right side is planar (Levine et al., 2017).
Skewness shows how the data is distributed and shows deviation (similar to mean and standard deviation). Skewness shows deviation assuming a normal distribution (distribution in the form of the bell curve where more frequent values occur near mean (central value) and frequency reduces moving away from the mean) (Holmes et al., 2017).
4 Median, Quartiles, and Box Plot
8 Median
Median refers to the middle value that separates the data set into two halves. While it serves a similar purpose as mean, however unlike means, it is not skewed by outliers (very small or very large value). Consider a data set 2,3,5,6,9. In this case, the mean is 5, and the median is 5. Now, if the number 9 is replaced by 14, the mean will become 6, the median will still be 5. Since it is a resistant statistic (not impacted by skewness), it is of high importance to ensure robustness (Siegel, 2012).
9 Quartile
A quartile is a statistic that divides the data set into four quarters. For the data to be divided into quarters, there is a need to arrange...