Analysis Of Call Center Performance Using Descriptive And Inferential Statistics

The Management Report: The aim of your report is to provide a meaningful interpretation of your data analysis to the business manager, demonstrating its use in decision making. Choose one discrete random variable and one continuous variable relevant to your topic. For example, a service provider may be interested in a discrete random variable counting the number of orders they receive per day, and a continuous random variable measuring the length of time it takes them to complete these orders.

To ensure a favorable response to your report, you need to explain to your manager any statistical terminology and techniques that you use in your report. Graphs may help the manager in understanding the concepts you are outlining. Rough work and calculations should not appear in the main body of the report but can be included in an appendix. As this is a management report, format your report appropriately. Marks will be deducted for unprofessional aspects of your typed report such as misspellings, poor grammar, unreferenced sources or inappropriate graphs

Describe what is a Discrete Probability Distribution. Identify/develop a Discrete Random Variable in your dataset. Use an appropriate Discrete Probability Distribution you have learnt in class to illustrate how you could apply these technique to solve your business problems.

Inferential Statistics

You may want to include some theoretical background to the Normal (and Sampling) Distribution. Topics could include:

What is the Normal Distribution?

Why is the Normal Distribution used in Sampling Distributions?

iii. What is the basis for Inferential Statistics?

1. You should include:
2. An explanation of your continuous random variable and sampling work
3. An explanation of what a Confidence Interval / Hypothesis Test is;

iii. An interpretation of your Confidence Interval/ Hypothesis Test results;

Question 1: Summary of Call Duration for Sample

The given report pertains to a call centre which has operations in multiple geographies i.e. Dublin, Cork and Galway. With regards to call centre, one of the parameters of operational performance of the staff corresponds to the number of hours received per hour and the proportion of unresolved calls. Typically, the calls received should be higher as this would lead to lower employee requirement and the proportion of unresolved calls should be low so as to ensure that the customer satisfaction is high. Additionally, for sample employees the call duration of 10 sample calls has also been recorded. Finally, the proportion of different types of calls attended by the randomly selected sample employees has also been summarised in a tabular manner by considering 10 random calls.

In wake of the above mentioned sample data, the objective is to present an analysis of the average call duration using various descriptive and inferential statistics tools and techniques. It is imperative to note that the objective of descriptive statistics use is to summarise the key characteristics of the sample data such as central tendency and dispersion in the sample data. The objective of inferential statistics usage is to estimate the population parameter using the sample data. For the purposes of the given report, confidence interval of mean call duration has been estimated for the call centre as a whole. Also, the potential relationship that exists between the call duration and the number of calls has also been explored using regression analysis as the tool of choice.

3. Analysis

The various tasks pertaining to the given data are performed in this section.

Question 1

Through this task, the aim is to present a summary of the average call duration for the sample. The concerned data corresponds to the call duration for 10 calls for the randomly selected 50 agents. As a result, the total sample size amounts to 500 calls. The task required a continuous variable and this variable is continuous in nature as the call duration can also assume non-integral values as the duration of calls need not be in integers.

The central tendency of the data is captured through the use of various descriptive summary statistics such as mean, median. The mean represents the average of the sample. The median value represents the mid-point whereby 50% of the sample values are lower than or equal to the median value. The average has been computed by taking into consideration the data for 50 agents and has come out as 5.49 minutes. This implies that on average it can be expected that a given customer call at the call centre would last for this much duration. The median call duration has come out as 5.51 minutes. This indicates that 250 customer calls from the given sample data would have call duration that would have lasted for less than or equal to 5.51 minutes (Eriksson and Kovalainen, 2015).

Also, the dispersion of the given variable needs to be expressed for which measures such as standard deviation, variance or Interquartile range (IQR) may be considered. For the given sample, the respective values of the above three statistics have been computed using Excel and summarised in the table below.

Question 2: Discrete Probability Distribution

The IQR value is 2.82 minutes which implies that middle 50% of the call duration would lie within duration of 2.82 minutes. Further, standard deviation for the call duration is 2.09 minutes which when compared to the mean clearly implies that there is moderate to high variation in the call duration in the sample data which is on expected lines as there depending on the underlying query of the customer and the underlying understanding ability of the customer and the agent, the call duration would show variation from the average time (Hair et. al., 2015).

The skew and kurtosis for the data have also been computed using Excel and are summarised below.

It is apparent that skew of the data is 0.11 which is quite low considering that 0 highlights the normal distribution which is perfectly symmetric. However, for the given distribution, the rightward tail would be marginally longer than the leftward tail. Also, the kurtosis is marginally negative and is close to zero which is the value required for normal distribution. The height and thickness of the peak is represented by kurtosis. If the kurtosis value is high in magnitude, then the peak is also taller while a lower magnitude indicates lower height. Further, the sign indicates the underlying thickness of the peak (Hillier, 2016).

The call duration distribution can be expressed in the form of following histogram

It is apparent from the above histogram that the distribution does seem to be symmetric and there are very few calls having duration which is more than 10.5 minutes. Further, it is evident that about 65% of the sample calls tend to range between 3 minutes and 7.5 minutes. Additionally, the above distribution tends to resemble a normal distribution quite closely considering that the histogram above would fit within a bell curve (Flick, 2015).

Question 2

 A discrete probability distribution tends to describe the underlying probability associated with the occurrence of all possible values of a random discrete variable. It is imperative to note that since the underlying variable is discrete, hence the values assumed would only be integral and corresponding probability of these would be captured.  Based on the given call centre data, a suitable discrete random variable of choice would be the number of calls that would be received in any given hour. This is a discrete variable as the number of calls received during an hour could be integer only and non-integral values cannot be assumed (Hastie,Tibshirani and Friedman, 2014).

The selected discrete variable probability distribution would be termed as Poisson distribution. This is a preferable to binomial distribution which is more useful when there are two possible outcomes one of which indicates a success and the other a failure. This is not the case here and instead the average calls received during an hour is known based on which the underlying probability of receiving n number of calls during the hour may be estimated. Based on this, the underlying probabilities may be determined (Medhi, 2016).

The respective probability distribution of the calls per hour from the sample data is indicated below.

Inferential Statistics

The respective probability density function corresponding to the above probability distribution is shown below.

Question 3

The normal distribution refers to density curve which is symmetric and bell shaped. Also, this distribution is defined by two parameters namely mean and standard deviation. The mathematical expression of the density curve of the normal distribution is represented as shown below (Lieberman et al., 2013).

In the above expression, the mean and standard deviation are captured by µ and σ respectively. Further, the median, mode and mean tend to coincide in a normal distribution and also the underlying skew is zero. The importance of the normal distribution can be explained with the help of Central Limit Theorem. In accordance with the Central Limit Theorem, the samples having a size of atleast 30 observations, it can be inferred that these are normally distributed (Hillier, 2016). Further, if the underlying population parameters are known, then the sample statistics may also be defined. For instance, the point estimate of the sample mean would be same as the population mean while the estimated sample standard deviation would be the population standard deviation divided by square root of the sample size. Using the Central Limit Theorem (CLT), it is possible to analyse the various samples and determine the underlying probability of an event occurring taking the underlying normal distribution and relevant characteristics into consideration (Lind, Marchal and Wathen, 2012).

Inferential statistics refer to the tools and techniques that are used to determine the population parameter by relying on the sample data. The underlying premise on which inferential statistics is based is that the sample has been selected randomly and is reflective of the underlying population. Only under the above assumption can reliable estimate about the population parameter be drawn (Flick, 2015). While, there are various techniques that are deployed in inferential statistics, for the purpose of the given data, confidence interval would be used.

Confidence interval refers to the technique whereby a particular parameter is estimated with a given level of confidence using the sample data. The higher the precision desired in estimation of confidence interval, the higher would be the confidence level chosen. For higher precision, a wider confidence interval would be framed while for lower precision, the confidence interval would be rather narrower. The confidence level would estimate the underlying probability associated with finding the requisite population estimate in the interval estimated using the sample data (Medhi, 2016). For the given exercise, the requisite continuous variable of choice is the call duration which is a continuous variable considering that it can assume non-integer values. The estimation of the 95% confidence interval with regards to the average call duration for the given call centre is as shown below.

Average sample call duration of staffs = 5.493 minutes

Standard deviation of Average sample call duration = 2.0946 minutes

Sample size (number of staff members in sample) = 500

Standard error = standard deviation / sqrt(sample size) = 2.09/ sqrt(500) = 0.09367

Degree of freedom = 500-1 =499

The t value for 95% confidence interval = 1.96473

Margin of error = t value * standard error = 1.96473*0.09367= 0.18405

Lower limit of 95% confidence interval = Mean – Margin of error = 5.493 -0.18405=5.31 minutes

Upper limit of 95% confidence interval = Mean + Margin of error =5.493 +0.18405= 5.68 minutes

From the above computation, it is apparent that the 95% confidence interval for the average duration of all calls made at the call centre would be [5.31,5.68] minutes. This implies that there is a 95% likelihood that the average duration of all calls made at the call centre would lie between 5.31 minutes and 5.68 minutes (Koch, 2013).

Question 4

The objective of this task is to conduct a linear regression analysis with the underlying independent variable being average call duration in minutes and the dependent variable being total hourly calls. It is expected that both the variables may be inter-related considering the fact that higher call duration would imply lower number of calls during any given hour. Additionally, both the given variables tend to have numerical values with underlying ratio scale of measurement which makes regression analysis a suitable tool (Eriksson and Kovalainen, 2015). The regression analysis for the chosen variables has been performed in Excel and the relevant output is shared below.

The estimated regression equation is as estimated below.

Number of hourly calls = 55.99 -1.14*Average duration of calls in minutes

In the above regression equation, value of 55.99 indicates the intercept while -1.14 is the slope of the regression line. The intercept value of 55.99 highlight that when the average call duration is zero, then 55.99 calls can be made within one hour. Further, as the duration of call increases by 1 minute, it would be estimated that number of calls made during the hour would reduce by 1.14 (Lieberman et. al., 2013).

Also, it is evident that the slope coefficient of the regression line is not significant considering that the p value is 0.33 implying the insignificance of slope at 99% confidence level. Besides, the coefficient of determination is also quite dismal as is evident from a value of 0.02 which implies that only 2% of the variation in the dependent variable can be accounted for by corresponding variation in the independent variable. This clearly reflects that the given model is not a good fit and therefore the current independent variable needs to be substituted with other independent variables (predictors) so as to improve the utility of the model (Hair et. al., 2015).

One of the key assumptions of a linear regression model is that homoscedasticity of the variables used in regression. This can essentially be tested using the residual plot of the error term. For the condition to be fulfilled, it is essential that the distribution of the error term in the residual plot should be random and hence no particular pattern should be observed (Hastie, Tibshirani and Friedman, 2014). For the given regression model, the residual plot is as indicated below.

It is apparent from the residual plot shown above that the residuals are indeed randomly placed with no particular pattern in sight and hence it may be concluded that the homoscedasticity assumption is fulfilled for the given regression model. The regression model may be used to estimate the number of calls that can be made within an hour for any given average call duration provided the value of the independent variable lies within the values used for the estimation of the regression analysis. Taking independent values which lie outside the range of inputs value of independent variable may result in unreliable estimate of dependent variable based on the regression analysis (Medhi, 2016).

Conclusion

In the above analysis, descriptive statistics have been used to carry out an analysis of the call duration whereby the various measures of central tendency, measures of variation coupled with skew and kurtosis have been used. The mean duration of each sample call comes at around 5.5 minutes with the median not being any significantly different from the mean value. The deviation in the data is moderately high which is expected considering that the call duration would depend on the nature of query and also the understanding capacity of the customer. The skew and kurtosis for the call duration is close to zero which is also reflected in the histogram thereby indicating that call duration is normally distributed.

With regards to discrete probability distribution, the suitable variable selected is number of calls received per hour since the data cannot be non-integral. The suitable probability distribution has been identified as Poisson distribution considering that binomial is not a suitable choice. Further, inferential statistics in the form of confidence interval has been used to estimate the mean call duration for all the calls made at the call centre using the sample data provided. Besides, the role of normal distribution and the underlying concept of normal distribution has also been discussed in some detail. Finally, regression analysis has been performed with call duration being thee independent variable and the number of hourly call being the dependent variable. The regression output does not indicate any significant relationship between the two variables which is also supported by the coefficient of determination which is close to zero. Also, the residual plot for the regression indicates that the underlying assumption of homoscedasticity has been fulfilled.

References

Eriksson, P. and Kovalainen, A. (2015) Quantitative methods in business research. 3rd ed. London: Sage Publications.

Flick, U. (2015) Introducing research methodology: A beginner's guide to doing a research project. 4th ed. New York: Sage Publications.

Hair, J. F., Wolfinbarger, M., Money, A. H., Samouel, P., and Page, M. J. (2015) Essentials of business research methods. 2nd ed. New York: Routledge.

Hastie, T., Tibshirani, R. and Friedman, J. (2014) The Elements of Statistical Learning. 4th ed. New York: Springer Publications.

Hillier, F. (2016) Introduction to Operations Research. 6th ed. New York: McGraw Hill Publications.

Koch, K.R. (2013) Parameter Estimation and Hypothesis Testing in Linear Models. 2nd ed. London: Springer Science & Business Media.

Lieberman, F. J., Nag, B., Hiller, F.S. and Basu, P. (2013) Introduction To Operations Research. 5th ed. New Delhi: Tata McGraw Hill Publishers.

Lind, A.D., Marchal, G.W. and Wathen, A.S. (2012) Statistical Techniques in Business and Economics. 15th ed. New York : McGraw-Hill/Irwin.

Medhi, J. (2016) Statistical Methods: An Introductory Text. 4th ed. Sydney: New Age International.