Applying Statistical Processes In Business Planning

Analyzing Quantitative Data

Our company is planning to enhance the decision-making and the information management through statistical processes acting as a research analyst. We are needed to demonstrate our understanding by applying statistical techniques in business planning with proper charts and tables. We are applying analytical statistical method for business planning. Then, we are communicating its findings. Generally, we would like to analyze quantitative data rather than qualitative data. Quantitative data provides more specific results of descriptive statistics, correlation, hypothesis-testing and regression analysis than qualitative data. 

Applying statistical processes to a number of areas of business planning and operations management is involved in inventory management and capacity management.

Variability refers how the data is “spread out” or scattered. The concept is synonymous to “dispersion”. The well-known measures of variability are range, inter-quartile range, variance and standard deviation.

Range:- The range is the easiest concept of measure of variability. In a business management dataset, the range is defined as deviation of minimum value from the maximum value.

Inter-Quartile Range:- The inter-quartile range (IQR) is the range of middle 50% of the frequencies of a distribution or a dataset (Beekman 2017). It is computed as the difference of 1^st and 3^rd quartile values. The measure of variability is needed to execute as it hampers the quality of products and materials in business sector. The variability brings more scatterness in the performance of business field. As example, we can check the proper frequency measures of month wise profits of any hotel. From this, we can measure the interval in which 50% of the sorted profits of hotel lie.

Variance:- Variability could also be stated as how close the frequencies in the distribution are the centre of the distribution. Utilizing the mean as the measure of the centre of the distribution, the variance is stated as the mean squared differences of the frequencies from the average (Dixon and Frank 1995). The variance helps to measure the variability of share and retail prices of any hospitality and event management organization. 

Standard Deviation:- The standard deviation is the square root of the variance. Standard deviation summarizes the quantity by which each value of the dataset differs from the mean. It infers how closely the values in the data sample are clustered around the mean with efficacy. It is the most robust measure of variability. Range and inter-quartile range do not takes into account all values of the dataset. However, standard deviation is calculated with respect to every variable in the dataset. More clustering refers lesser standard deviation and vice versa. Standard deviation is commonly represented in collaboration with mean and measured in the same unit of the data. As an example, the payment level fluctuation in a Hospital or Nursing home of the employees, doctors of nurses are measured by standard deviation.

Measures of Variability

Variance of analysis includes the assessment of difference of variance or mean. It aims to determine the cause of variability. Variance analysis maintains control over the expense of project by monitoring planned versus actual costs. Effective variance analysis could help an organization to spot trends, challenges, scopes and threats for short or long duration success. Variance analysis estimates and predicts various types of aspects such as “Budget vs. Actual Cost”, “Materiality”, “Relationships” and “Forecasting”.

Variability is crucial for better understanding of the customer experience. The presence of smoothing effects is highly variable. The target of business authorities is to decrease variability so that the customer could have confidence in the delivery estimates of business authority. The reduced variability can help the company delivery estimates and pinpoint enhancement scopes. The variability tree of performance indicates predictable and unpredictable variability in market share of business. Managing variability is the process that takes work and knowledge of tools that are helpful and pragmatic. The variability in business sector helps business employees to understand the customer better, understand the improvement and identify the ways in which we could further delight the customer.

The variability in business processes creates a significant issue. Variation is the enemy of quality. The variation could cause due to inevitable common causes and assignable causes. Process enhancement activities could decrease inevitable causes by but it could not be eliminated. The variation is inherent in a process that is functioning as designed. However, variation of assignable causes is unnatural variation in a method. It could be detected and rectified. Control charts helps indetermination of the type of variable that exists in a process.

A common challenge in business is managers who consider all variation as if it were due to assignable causes. That leads to the over-adjustment of the methods that enhances variation, criticism of employees for variation that is out of control. A quality engineer notices a pattern on the X-bar chart of a turning method.

Statistical process control (SPC) is applicable for statistical techniques for controlling a process and quality. Statistical Process Control is divided into generation of control chart and process capability study. Control charts give a means of determining the type of variation that is present in a process. Process capability study executes the ability of the “in control” method to generate product that attends specifications.

The six-sigma process quantifies the statistical process performance and relates quality management. As the method sigma value increases from zero to six, the variations of the method around mean value reduces. When the value of sigma is high enough, the process approaches to null variation and is known as “zero defects” (Rigdon and Woodall 2017). The aim of our company is to restrict the defects of lot and reach to perfection by minimum variability. In statistical process control of quality management, standard deviation becomes mediator between mean of the process and specification as well as control limit. The Control limits that are Upper Control Limit (UCL) (µ+3*σ) and Lower Control Limit (µ-3*σ) detects the defective raw elements and materials in the lot of products (Brue, 2002). The specification limit that are Upper specification limit (µ+6*σ) and Lower specification limit (µ-6*σ) gives better relaxation than control limits to the defective elements (Page 1995). Generally, we denote Upper and Lower Control Limits as six sigma limit as the distance of these two classification lines are measured 6σ (six-sigma). It is noted that almost 99.73% product of a lot lies in the interval of specification limits and 73.1% product of a lot lies in the range of Control Limits (Pande, Neuman and Cavanagh 2000). The performance is worse than the foresaid criteria indicating a bad lot of product or material. We must summarily reject the lot due to severe vulnerability of customer satisfaction (Barone and Franco 2012).

Variance Analysis

The statistical process control method has lasted for decades as it got effective result. Companies and industries in every country around the world have launched logical and systematic process control method to improve the quality, services and products of the processes. Statistical process control includes an expert deal of statistics and data analysis. Statistical aspects often stimulate individuals, who are eager to enhance and management quality. Data analysis is a critical component of quality improvement. Statistical quality control is a systematic, data-driven approach and methodology used to eliminate defective items in a lot of product (Kwak and Anbari 2006). From manufacturing to selling, in every types of product services Six-Sigma concept is being used. The graphing representation of Statistical process control is control charts. A six-sigma defect is measured as any product lying beyond the control limits of control charts. Statistical Process Control manages the accomplishment of definition, measurement, analysis, improvement and control (Saraph, Benson and Schroeder 1989).

The control-charts (Zhang et al. 2014)

Name of control charts	Observation of processes	Types of observations of process
Mean and Range chart (X-bar and R chart)	Characteristics are measured within one subgroup	Variables
Mean and Standard deviation chart (X-bar and S chart)	Characteristics are measured within one subgroup	Variables
Moving Bar chart or Moving Range chart	Characteristics are measured from individual sample	Variables
p-chart	Fraction nonconforming within one subgroup	Attributes
np-chart	Fraction nonconforming within one subgroup	Attributes
c-chart	Fraction nonconforming within one subgroup	Attributes
u-chart	Fraction nonconforming within one subgroup	Attributes

Probability distributions are an integral part of statistical theory. In business, probability theory is used in the calculation of long-term profit and losses and many other works relevant to business. For example, a market is examining a sample of new food, new medicine, new electronic gadget, new drink, new automobile product or other new accessories. Director of production or Manager of Marketing a company makes the primary concept about outcome whether customers would prefer the product if it is manufactured and distributed in lot or not.

Generally, probability distributions could be divided in two types. These are discrete probability distribution and continuous probability distribution. In case of discrete probability distribution, the data is countable and countably finite in nature. Oppositely, continuous probability distributions are consisting of countably infinite number of data.

Discrete probability distribution is used for providing the information about possible coming orders with the help of Poisson distribution. Binomial distribution helps to measure frequency distributions of many binary or continuous data behavior. Hypergeometric or waiting time distribution helps in operational research and optimizing techniques of respective business field.

In probability and statistics, discrete probability distribution is known as a probability distribution of a function that has countable values and determines a probability mass function (f(x)). Hence, the probability distribution of a random variable X is discrete in nature. X is called a discrete random variable if and only if- 

Here, u belongs to X.

Managing Variability

The discrete probability distributions are dependent up on random variables and sample spaces. The well-known discrete probability distributions are Bernoulli distribution, binomial distribution, Poisson distribution, uniform distribution, geometric distribution, hyper-geometric distribution and negative binomial distribution.

A random variable X with the discrete uniform distribution on the integers i=1(1)m has probability mass function,   We write X~ distunif (m).

The binomial distribution is established with the help of Bernoulli trial. Bernoulli trial is a randomly conducted experiment in which there are only two possible outcomes: success (S) and failure (F). We incorporate the Bernoulli trial and suppose- 

If the probability of success is p then the probability of failure is q = (1-p) and the probability mass function of X is – 

Here, the mean is p and the variance is p*q. Therefore, the binomial probability is- 

Therefore, the PMF of X is- 

We say that X has a hypergeometric distribution and X ~ hyper (m=M, n=N, k=K).

Let, X be the number of failures before a success. If P(S)=p, then X has the probability mass function-      

Therefore, X ~ geom(prob = p). X follows a geometric distribution.

Let, X counts the number of failures before r successes. If P(S) = p, then X has probability mass function- 

Therefore, X ~ nbinom (size = r, prob = p). X follows a negative binomial distribution.

The Poisson distribution is used for structuring the number of times an event occurs in an interval of time and space. Here, an event occurs in the interval k-times and k =1(1) n. The occurrence of one event does not hamper the probability that next event would happen. Hence, events occur independently (Collins et al. 2015). The rate of event occurrence is constant for every interval. The possibility of an event in a small sub-interval is proportional to the length of sub-interval. Poisson distribution is actually extended version of binomial distribution when sample size is large.  

Let, λ be the average number of occurrence in the time interval [0,1]. Suppose, the random variable X count the number of events occurred in the interval. Then, it can be shown for a large sample space that- 

Here, X ~ pois (λ).

Poisson distribution is used for modeling queuing system, for example, people arriving at railway ticket counter or airport.

Continuous probability distributions are made up with continuous random variables. The probability that a continuous random variable would take a particular value is 0. As an outcome, a continuous probability distribution could not be expressed in tabular form. The equation utilized to derive continuous probability distribution is called probability density function. Here, y = f(x). The value of y ≥ 0, for all values of x. The popular continuous probability distributions are Beta distribution, Gamma distribution, chi-square distribution, F-distribution, t-distribution, exponential distribution, Weibull distribution, log-normal distribution and more of it normal distribution.

Statistical Process Control

In analog-to-digital conversation, quantization error is approximated by uniform distribution. If we are eager to find out the numbers of aircraft carriers produced by Emirates in Abu Dhabi per year, we should apply normal distribution. In the tele-marketing centers the majority of the people hang up immediately and others taper on longer. There, exponential model is followed. Log-normal distribution is utilized to model returns from stock market. The launching lag of new automobile sets of any company follows exponential distribution. In finance and risk-management, fat-tailed distributions such as t distribution are applied. These most appropriate distributions applied in those sectors are skewed distribution. To find the life-expectancy of any product, the business organizations relies up on weibull, exponential and gamma distributions. The pharmaceutical and druggist companies often use laplace, log-normal or gamma models for data analysis and decision-making. The estimating price fluctuation in terms of fall or hike of any machinery, the weibull distribution is applied. Queuing theory of business organizations is solved by Gamma distribution. The Chi-square distribution looks for the association among different non-parametric variables of stock-market or retail prices.

Normal distribution is also popular as Gaussian distribution that looks like a bell shaped curve. The probability density function of a Normal distribution is-

       (x is finite)

Here, µ is the mean and σ² is the variance of normal distribution (Feller 2015). Normal distribution is symmetric in nature. That is, its mean=median=mode. All the continuous and even some discrete distributions could be standardized to normal distribution by central limit theorem (Conrick and Hanson 2013).

In case of inferential statistics, testing of hypothesis is the main discussing part. Inferential statistics gives us the scope of better understanding and decision-making. The ingredients for making this calculation are the same for all statistical procedures:

1. The sample size
2. The variability of the sample
3. The size of the observed differences

Inferential statistics give a pathway from a “sample” to a “population” (Hacking 2014). This segment of statistics infers the parameters of a population from data on the statistics of a sample. It is generally essential for the researcher to deal with samples rather than an entire population. However, one issue is that a sample generally is not identical to the population from which it comes. Especially, the sample mean would differ from the population mean. That is-

X-bar ≠ µ

Standard error is a crucial measure. It specifies how good a sample mean estimates the population mean. Normality assumption is a prior criterion for inferential conclusion drawing of any distribution.

Probability Distributions

Hypothesis testing is an inferential process that utilizes sample data for evaluating the credibility of a hypothesis about a population. We construct the structure of null hypothesis and alternative hypothesis. We use necessary estimation and standard error procedure and reject or do not reject the null hypothesis. We, use the symbol S_x-bar that indicates the calculated values from sample data rather than from the population parameter. Here, S_x-bar = S/SQRT(n).

If the null hypothesis is true, then the following criteria will satisfy-

Determining margin of error (ME) at different confidence intervals are necessary tasks. Margin of error is the range of values above and below the sample statistic in confidence interval. It ensures us how many percentage points our outcome would deviate from real population value. As example, a 95% confidence interval with a 5% margin of error refers that the sample statistic would be within 5% confidence intervals of the real population with 95% probability (Fay 2017). The margin of error is calculated as-

Margin of error = Critical value * Standard deviation

Margin of error = Critical value * Standard error of the sample statistic.

The margin of error of a sample statistic for various proportions is given as-

Margin of error =

Here, p-hat = sample proportion, n = sample size, z= z-score.

The confidence interval is the way-out for determining what the uncertainty is with a certain statistic. The 95% confidence interval of population parameter of any event is if x and y, then we can say that the event has 95% probability to fall in the confidence interval x and y.

We apply various types of tests such as t-test, chi-square test, ANOVA test, F-test or other normality tests for analysis of data. The tests are based on error (type I error and type II error) and p-value concept. We generally assume the confidence intervals 90%, 95% of 99%. The p-values are correspondingly, 0.1, 0.05 and 0.01. We reject the null hypothesis if calculated p-value is less than assumed critical region (p-value = α = 0.05). We perform statistical tests to examine if obtained sample characteristics are sufficiently different from expected null hypothesis.

Relationships between pairs of variables may also be detected when variance analysis is performed. A probability analysis is a statistical function that detects all the conceivable outcomes that random variable would have within a controlled range. Probability distributions could be used to generate scenario analysis. A scenario analysis utilizes probability distributions various theoretical distinct probabilities for the result of the particular course of actions for quality management. Probability distribution refers the probable outcomes of a defined event’s likelihood function. It especially is true for smaller businesses that tend to have more volatility than larger organizations, newer businesses without a valid way of sales and costs. Therefore, probability distributions could be a great tool for estimating quality management components. Besides, risk evaluation and sales management is executed by probability distribution approach.

One of the most practical uses of probability distributions in business sector is to predict future levels of sales. Using a scenario analysis on the basis of a probability distribution could support company frame its probable values in terms of a likely level of sales and a worst-case and best-case scenario. In a circumstance of competitive business, the statistical strategies offered in probability analysis can indicate entrepreneurs the most likely results and most profitable ways. Probability analysis features formulas that business owners could employ in a restricted manner to anticipate potential outcomes. A major application for probability distribution depends in anticipating future sales incomes. Companies of all sizes depend on sales forecasts for predicting revenues. Probability distributions can support companies to neglect negative results for predicting positive outcomes. The scenario analysis helps to venture producing tools for evaluating different business scenarios. Probability analysis employs probability distributions to generate various business scenarios.  

The most effective way of communicating the outcomes of the result of our analysis and variables is inferential statistics and hypothesis-testing. Mainly, nominal (numeric) data uses this procedure. Nominal variables are countable and quantitative. It has mainly two levels that are interval and ratio levels. An interval scale measures the difference of two numerical variables that is justified. A ratio variable has the identifications of an interval variable that has a prominant definition of 0. For example, the differences between (100 degree Celsius – 90 degree Celsius) and (90 degree Celsius – 80 degree Celsius) is equal in interval scale. However, it is not equal in case of ratio scale. As degree Celsius is a numeric ratio scale. In business purpose, ratio scale is more used than interval scale of numerical variables (Freelon 2013).

Oppositely, nominal and ordinal scales are used for categorical variables. Examples of nominal scales in business sector are – “yes” and “no”, “profit” and “loss” etc. These are qualitative data scales (Zumbo 2014). Ordinal variable is also the categorical variable. However, in this scale order matters not the variability between values (Jelizarow, Mansmann and Goeman 2016). For example, “good”, “moderate” and “bad” are the levels of ordinal variables. “Strongly disagree”, “Disagree”, “Neither disagree nor agree”, “Agree” and “Strongly agree” are the “Likert” scale (ordinal scale) measure of the variable.

Computable Functions	Nominal scale variables	Ordinal scale variables	Interval scale variables	Ratio scale variables
Frequency distribution	yes	yes	yes	yes
Median, quartiles and percentiles	no	yes	yes	yes
Vector operations (addition, subtraction etc.)	no	no	yes	yes
Central tendencies (mean, mode,)	no	no	yes	yes
Dispersion functions (SD, variance, standard error of mean)	no	no	yes	yes
Ratio and Coefficient of variation	no	no	no	yes

We are aware of frequency distribution of the data. Based, on classified (grouped and ungrouped) data and frequency, we determine the appropriate table. For making a frequency distribution table for ungrouped data is easier than the grouped data. We are discussing different graphs such as bar graph, histogram, frequency polygon, pie charts, OGIVE and normal curve.  

Continuous frequency table indicates the class interval and class boundaries along with mid values for all the classes. Discrete frequency table indicates the discrete mark value. Now with the help of tally marks, we put the relevant frequencies to the classes or the discrete value. We can further find cumulative frequencies (from upper to lower or lower to upper), percentage of frequencies and cumulative percentage frequencies (from upper to lower or lower to upper) in the frequency table. From frequency table, we can draw necessary graphs and charts.

class interval	class boundary	class mid-value	frequency	cumulative frequency(uper to lower)	cumulative frequency(Lower to upper)	percentage of frequency
1-10	0.5- 10.5	5.5	8	8	50	16
11-20	10.5-20.5	15.5	10	18	42	20
21-30	20.5-30.5	25.5	12	30	32	24
31-40	30.5-40.5	35.5	7	37	20	14
41-50	40.5-50.5	45.5	13	50	13	26

The class boundary 41-50 has maximum frequency and percentage of frequency (13, 26%) followed by the class boundary 21-30 (12, 24%). Correspondingly, the class boundary 31-40 has minimum frequency (7, 14%).

In bar chart, bars of corresponding frequencies must be proportional to the quantities, they are representing. The width of bars must be equal and there must be uniform space between bars. Mainly, discrete variables are measured in various types of bar graphs such as column bar, stacked bar, row bar, grouped bar, percentage bar plots etc. Levels of values are presented in one axis and frequencies are represented in other axis.

The frequencies in the frequency table reflect the bar sizes in bar plot. The larger bar is with mid-value 45.5 and smaller bar is with mid-value 35.5. 

Histogram is a graph that looks like bar plots but the only difference is there is no gap between bars. Instead of class intervals or class mid value, we use class boundaries in histogram plot. It is formed by adjacent rectangles whose width is the distance from lower to the upper class boundary and the height is the relevant frequencies (Scott 1979). 

Frequency polygon is a specification of histogram plotting. For multiple frequency polygon, a legend would facilitate understanding of the information the graph wants to convey. The upper mid-points of histogram rectangles are connected to each other. It provides us the frequency polygon. It shows the association between two or more sets of continuous data. 

Pie chart is used while we compare all the parts in a whole. The size of each sector of the circle is proportional to the size of the category that it represents. In a circle, we construct successive central angles using the number of degrees representing each part (Simkin and Hastie 1987). The percentage of each part is proportional to the segment placed in a circle according to the angle. 

OGIVE chart is a frequency polygon where the cumulative frequency of each class is plotted against the corresponding class boundaries. For the same data, the less than OGIVE and the greater than OGIVE are graphed in one grid. The blue line in this OGIVE indicates lower to higher cumulative frequencies and the red dotted line indicates higher to lower cumulative frequencies for five class intervals. 

Normal curve is a fitted predictability pattern of data. From histograms, we can generate fitted normal curves. It is based on “empirical rule” of the data. It helps to find the nature and normality of data.

The curve line in the below normally fitted histogram indicates that the mean lies in frequency 10 with standard deviation 2.55. 

The advantage for frequency table is that, we can get summarized occurrences of several classes of a data in a single table. However, we cannot find the central tendency or sctterness of the classes. Bar plots give us the class wise bar representation but do not give the distribution of data. Frequency polygon and OGIVE provide more compact trend and distribution of data. However, it is only dependent upon mid-value. Therefore, bias comes. Similarly, histogram is a nice representation of data. However, it is not able to provide comparative study. Pie chart is a perfect chart for comparing class wise variations. However, pie chart could not interpret the nature of data in each class.

MS Excel, Minitab and SPSS are very vibrant software used by generation of charts, tables and graphs of a data.

References:-

Beekman, I., 2017. Algorithm for computing descriptive statistics for very large data sets and the exa-scale era. Bulletin of the American Physical Society.

Collins, D.J., Neild, A., Liu, A.Q. and Ai, Y., 2015. The Poisson distribution and beyond: methods for microfluidic droplet production and single cell encapsulation. Lab on a Chip, 15(17), pp.3439-3459.

Conrick, C. and Hanson, S., 2013. Normal Distribution, Probability, and Modern Financial Theory. Vertical Option Spreads: A Study of the 1.8 Standard Deviation Inflection Point, pp.93-109.

Dixon, W.J. and Massey Frank, J., 1995. Introduction To Statistical Analsis. McGraw-Hill Book Company, Inc; New York.

Fay, Michael P.2017.”exact2x2: Exact Conditional Tests and Matching Confidence Intervals for 2 by 2 Tables.”

Feller, W., 2015. On the normal approximation to the binomial distribution. In Selected Papers I (pp. 655-665). Springer International Publishing.

Freelon, D., 2013. ReCal OIR: Ordinal, Interval, and Ratio Intercoder Reliability as a Web Service. International Journal of Internet Science, 8(1).

Hacking, Ian., 2016. Logic of statistical inference. Cambridge University Press.

Holcomb, Z.C., 2016. Fundamentals of descriptive statistics. Routledge.

Jelizarow, M., Mansmann, U. and Goeman, J.J., 2016. A Cochran–Armitage?type and a score?free global test for multivariate ordinal data. Statistics in medicine, 35(16), pp.2754-2769.

Mitra, A., 2016. Fundamentals of quality control and improvement. John Wiley & Sons.

Rigdon, S.E. and Woodall, W.H., 2017. Using the predictive distribution to determine control limits for the Bayesian MEWMA chart. Communications in Statistics-Simulation and Computation, pp.1-9.

Scott, D.W., 1979. On optimal and data-based histograms. Biometrika, 66(3), pp.605-610.

Simkin, D. and Hastie, R., 1987. An information-processing analysis of graph perception. Journal of the American Statistical Association, 82(398), pp.454-465.

Zhang, Y., He, Z., Zhang, C. and Woodall, W.H., 2014. Control charts for monitoring linear profiles with within?profile correlation using Gaussian process models. Quality and Reliability Engineering International, 30(4), pp.487-501.

Zumbo, B.D., 2014. Nominal Scales. In Encyclopedia of Quality of Life and Well-Being Research (pp. 4368-4369). Springer Netherlands.