Statistics of Employees - Case Study Example

Executive Summary This paper employees various statistical tools to study information on 474 employees working for a large organization in Wales. It emerged that most employees are clerks while those working in administration, technical and security make up 8%, 7%, and 6% respectively of the total employees. In the same line, only 2% of the employees are executive members. The distribution of beginning salary of Apple’s employees is positively skewed with most people earning low salaries while few individuals earn extremely high beginning salary. Additionally, there is a negative and weak relationship between salary now and age given a correlation of However a correlation coefficient of -0.0096 between beginning salary and age paints a picture that the two variables do not have any relationship. Further analysis of salary now and salary at the beginning shows that the variables are positively related. In conclusion, gender of employees in the company and educational level are interdependent such that male in the company tend to hold higher educational qualifications compared with their female counterparts. Introduction The first task is to summarize the distribution of the data using pie charts, bar graphs, and frequency distribution curves. In a bid to understand the shape of the distribution, histograms and polygons will be constructed. The second section of the paper will comprise construction of scatterplots and coefficients of correlation between various pairs of variables. The third task is to utilize probability to draw inferences on interdependence between Educational Level and Sex in the company. Lastly, the concept of normal distribution will be employed in finding the probability now of earning at least £15,000 for males as well as for females. Task 1: Summarizing Data 1. The table below gives the classification of each variable and states whether the variable is quantitative or qualitative (Glaser, 2005). Variable Classification Quantitative or Qualitative IDNUM Ordinal scale Categorical variable SALBEG Ratio scale Quantitative variable , it is possible to measure and order proportions SEX Nominal scale because they are discrete in nature Qualitative variable AGE Ratio scale Quantitative variable , it is possible to measure and order proportions SALNOW Ratio scale Quantitative variable , it is possible to measure and order proportions JOBCAT Nominal scale because they are discrete in nature Qualitative variable ISWELSH Nominal scale because they are discrete in nature Qualitative variable EDLEVEL Ordinal scale Categorical variable AGEBAND Interval scale Quantitative variable, it is possible to measure and order proportions SALNBAND Interval scale Quantitative variable, it is possible to measure and order proportions 2. Presenting Job Category as a Pie Chart and as a Simple Bar Chart According to pie chart shown below, 48% of employees are clerks while 2% are executive. This means that the biggest proportion of employees work as clerks. While trainees make up 29% of Apple’s workforce, those working in administration, technical and security make up 8%, 7%, and 6% respectively of the total employees. pie chart of job category The bar graph below supports the assertion that most employees work as clerks while a very small proportion work as executives. This is displayed by the height of the bars. Figure 2: Bar graph of job category 3. Grouped frequency distribution and cumulative frequency distribution of Salbeg data 4. Histogram, Frequency Polygon and a Relative Cumulative Frequency curve for Salbeg data are shown below. Histogram of Salbeg The histogram has a longer tail stretching to the right with the peak located to the left side. The shape of the distribution is an indicator that a few individuals have extremely high beginning salary effectively pulling the distribution in the rightward direction (Black, 2011). The frequency polygon below supports further the presumption that few individuals earn extremely high beginning salary effectively pulling the distribution rightwards. Frequency polygon of Salbeg LeBlanc (2004) asserts that Relative Cumulative Frequency curve is constructed after the application of the relative frequency formula given as: According to the cumulative relative frequency curve, 100% of the employees earn a beginning salary that is less than 35,000. Similarly, approximately 80% of the employees earn below $8,000. 5. Using Relative Cumulative Frequency curve to estimate the median and quartiles of Salbeg The median is obtained by finding the position on the vertical axis corresponding to 0.5. A horizontal line is drawn to touch the curve and then drops to touch the x-axis. The corresponding value is estimated as $6000. The same procedure is followed when examining the first and third quartile. It is visible from the graph above that the first and third quartile is $5,000 and $7,000 respectively. 6. The variable Salbeg is positively skewed with a long tail stretching to the right. Because of the skewness an appropriate measure of central tendency for the distribution is the median. In a skewed data, mean is usually dragged in the direction of skewness hence median ought to be used because it is unaffected by extreme values. The median earning is $6,000. Besides being positively skewed, the distribution is unimodal with a single mode. This is evidenced by a single pillar. Task 2: The regression and correlation 1. Scatterplots and coefficients of correlation (a) Salary Now and Age Using excel function =CORREL(A2:A475,B2:B475), salary now and age have a correlation of This is a negative relationship which shows that as age increase, salary now tends to decline. However, the relationship is weak. By observing the slope of the line fit salary now reduces as age of employees increase. Concisely, each improvement in age leads to low current salary. (b) Salary at Beginning and Age The coefficient of correlation shows a very weak and negative relationship between age and the beginning salary. Since the coefficient of correlation is approaching zero, and that the line plot is nearly horizontal, the appropriate conclusion is that age and beginning salary are not related in any way. (c) Salary Now and Salary at Beginning. The calculated coefficient of correlation value of indicates a strong positive relationship between salary now and salary at the beginning. Concisely, salary now increases as the beginning salary increase. 2. Regression equation of Salary Now on Salary at Beginning In order to obtain a regression equation of Salary Now on Salary at the Beginning, it is important to utilize summary regression output calculated using excel. The result has been tabulated in the table below. There exists a relationship between current salary and the beginning salary. This relationship is depicted by the positively sloping line fit plot obtained from regression analysis and illustrated using scatter plot above. According to the regression equation, each improvement in salbeg leads to higher salnow. In this case, higher current salary is associated with higher beginning salary. The regression equation is as follows: The interpretation of the regression equation is that salnow increase by 1.8928 for each increase in beginning salary, ceteris paribus. At the same time, salary now for an employee would be 1125.4 if the beginning salary were zero, other factors held constant. According to the line of best fit drawn above and the Beginning Salary of $20,000, the predicted Salary Now would be approximately. The regression equation can also be used to predict the Salary Now as follows: The accuracy and reliability of the estimates of salary now calculated above is determined by looking coefficient of determination of the regression model. The coefficient of determination, R-square value is 0.76 and shows that 76% of the variability in dependent variable is explained by the independent variable. Task 3: Probability 1. A pivot table of Sex and Educational Level By using excel, the pivot table of Sex and Educational Level is tabulated below. An improved pivot table of Sex and Educational Level is tabulated with titles is illustrated in the table. 2. The probabilities that the selected staff is: (a) In the professional qualification category There is a 4.22% chance that the selected staff has professional qualifications. (b) In the no formal qualification category There is a 11.18% chance that the selected staff has no formal qualifications. (c) Female There is a 45.57% chance that the selected staff is a female. (d) Male There is a 54.43% chance that the selected staff is a male. (e) In the Professional Qualification category given that they are female. (f) In the Professional Qualification category given that they are male. (g) In the No Formal Qualification given that they are female There is a 13.89% chance that the selected staff has no formal qualification given that she is a female. (h) In the No Formal Qualification category given that they are male There is a 8.91% chance that the selected staff has no formal qualification given that he is a male. In conclusion male in the company tend to hold higher educational qualifications compared to female. 3. In order to test whether Educational Level is independent of Sex, this paper will utilize professional qualification category and female sex in the analysis. The process will involve establishing whether the following equation holds: By using the pivot table; The values calculated above are then used to test if the presented equation holds; Since , it follows that The conclusion therefore is that Educational Level is not independent of Sex. In this case, gender of employees in the company has a determining factor on educational level. Task 4: Probability distribution 1. Continuous variables that are available in the file are Salbeg, Age, and Salnow. These variables are continuous because they can take any value between its minimum and maximum. 2. If the company wants to establish a new office with eight occupants using the proportion of makes to female and assuming an independent random selection of each office member, the probability distribution for the number of females in the office is obtained using binomial probability distribution function tabulated as follows The corresponding excel function is: The corresponding probability distribution graph for the number of females in the office is shown below. 3. The first assumption is that the population has to be normally distributed since we are not given the population mean but the sample size is large i.e. 258 males and 216 females. Under the assumptions of normal distribution, sampling distribution of the mean is normal when the distribution of population mean is normal (Heiman, 2011). In other words, the distribution of means across samples is normal for a normally distributed population. It further implies that for a normal distribution with mean, the distribution of the sample mean is normal with a mean of . (Weiers, 2010). This assumption is relevant to be able to evaluate the probability of earning at least $15,000 for males and females. The tabulated descriptive statistics pertinent for the assessment of probability Using the descriptive statistics above, the probability now of earning being at least $15,000 for male is: The calculated z-score is negative effectively showing that the stated earnings $15,000 lies below the average earning. The probability implies that 58.06% of male currently earn above 15,000. This is approximately 150 male employees. The result is in agreement with personnel manager’s assertions that most male employees will eventual earn above 15,000. In the case of female, the probability now of earning being at least $15,000 is: The probability calculated above means that 7% of female employees currently earn above 15,000. This is approximately 15 female employees. The calculated value is substantially below the 50% benchmark that female employees will eventually earn above 15,000. Currently, only 7% of female employees earn above 15,000. Conclusion One of the conclusions is that most employees are clerks while only 2% of the employees are executive members. Secondly, the distribution of beginning salary of employees is positively skewed with most people earning low salaries while few individuals earn extremely high beginning salary. While the relationship between salary now and age is negative, salary now and salary at the beginning are positively related. The paper also showed that 58.06% of male and 7% of female currently earn above $15,000. Finally, sex and educational attainment are interdependent and that male in the company tend to hold higher educational qualifications compared to female. Reference List Black, K 2011, Business Statistics: For Contemporary Decision Making, Somerset, NJ, John Wiley & Sons. Glaser, G 2005, High-yield Biostatistics, Philadelphia, Lippincott Williams & Wilkins. Heiman, G 2011, Behavioral Sciences STAT. Mason, OH, Cengange Learning. LeBlanc, D 2004, Statistics: Concepts and Applications for Science, Burlington, Jones & Bartlett Learning. Weiers, R 2010, Introduction to Business Statistics, Mason, OH , Cengage Learning. Read More