Master of Business Administration

Semester 1

STATISTICS FOR MANAGEMENT

Set- 1

1. (a) ‘Statistics is the backbone of decision-making’. Comment

(b) ‘Statistics is as good as the user’. Comment

Ans.

(a) ‘Statistics is the backbone of decision-making’

Due to advanced communication network, rapid changes in consumer behavior, varied expectations of variety of consumers and new market openings, modern managers have a difficult task of making quick and appropriate decisions.

Therefore, there is a need for them to depend more upon quantitative techniques like mathematical models, statistics, operations research and econometrics.

Decision making is a key part of our day-to-day life. Even when we wish to purchase a television, we like to know the price, quality, durability, and maintainability of various brands and models before buying one. As you can see, in this scenario we are collecting data and making an optimum decision. In other words, we are using Statistics.

Again, suppose a company wishes to introduce a new product, it has to collect data on market potential, consumer likings, availability of raw materials, feasibility of producing the product. Hence, data collection is the back-bone of any decision making process.

Many organizations find themselves data-rich but poor in drawing information from it.

Therefore, it is important to develop the ability to extract meaningful information from raw data to make better decisions. Statistics play an important role in this aspect.

Statistics is broadly divided into two main categories. Below Figure illustrates the two categories. The two categories of Statistics are descriptive statistics and inferential statistics.

•Descriptive Statistics:

Descriptive statistics is used to present the general description of data which is summarized quantitatively. This is mostly useful in clinical research, when communicating the results of experiments.

•Inferential Statistics:

Inferential statistics is used to make valid inferences from the data which are helpful in effective decision making for managers or professionals.

Statistical methods such as estimation, prediction and hypothesis testing belong to inferential statistics. The researchers make deductions or conclusions from the collected data samples regarding the characteristics of large population from which the samples are taken.

So, we can say ‘Statistics is the backbone of decision-making’.

(b) ‘Statistics is as good as the user’

Statistics is used for various purposes. It is used to simplify mass data and to make comparisons easier. It is also used to bring out trends and tendencies in the data as well as the hidden relations between variables. All this helps to make decision making much easier. Let us look at each function of Statistics in detail.

1. Statistics simplifies mass data

The use of statistical concepts helps in simplification of complex data. Using statistical concepts, the managers can make decisions more easily. The statistical methods help in reducing the

complexity of the data and consequently in the understanding of any huge mass of data.

2. Statistics makes comparison easier

Without using statistical methods and concepts, collection of data and comparison cannot be done easily. Statistics helps us to compare data collected from different sources. Grand totals, measures of central tendency, measures of dispersion, graphs and diagrams, coefficient of correlation all provide ample scopes for comparison.

3. Statistics brings out trends and tendencies in the data

After data is collected, it is easy to analyze the trend and tendencies in the data by using the various concepts of Statistics.

4. Statistics brings out the hidden relations between variables

Statistical analysis helps in drawing inferences on data. Statistical analysis brings out the hidden relations between variables.

5. Decision making power becomes easier

With the proper application of Statistics and statistical software packages on the collected data, managers can take effective decisions, which can increase the profits in a business.

Seeing all these functionality we can say ‘Statistics is as good as the user’.

2. Distinguish between the following with example.

(a) Inclusive and Exclusive limits.(b) Continuous and discrete data.(c) Qualitative and Quantitative data(d) Class limits and class intervals.

Ans.

(a) Inclusive and Exclusive limits.

Inclusive and exclusive limits are relevant from data tabulation and class intervals point of view. Inclusive series is the one which doesn't consider the upper limit, for example, 00-10 10-20 20-30 30-40 40-50 

In the first one (00-10), we will consider numbers from 00 to 9.99 only. And 10 will be considered in 10-20. So this is known as inclusive series. Exclusive series is the one which has both the limits included, for example, 00-09 10-19 20-29 30-39 40-49 

Here, both 00 and 09 will come under the first one (00-09). And 10 will come under the next one. 

(b) Continuous and discrete data.

All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get 0, 1, 2, 3, etc. 

All data that are the result of measuring are quantitative continuous data assuming that we can

measure accurately. Measuring angles in radians might result in the numbers p/6, p/3, p/2, p/, 3p/4, etc. If you and your friends carry backpacks with books in them to school, the numbers of books in the backpacks are discrete data and the weights of the backpacks are continuous data. 

(c) Qualitative and Quantitative data:

Data may come from a population or from a sample. Small letters like x or y generally are used to represent data values. Most data can be put into the following categories: 

• Qualitative • Quantitative 

Qualitative data Qualitative data are the result of categorizing or describing attributes of a population. Hair color, blood type, ethnic group, the car a person drives, and the street a person lives on are examples of qualitative data. Qualitative data are generally described by words or letters. For instance, hair color might be black, dark brown, light brown, blonde, gray, or red. Blood type might be AB+, O-, or B+. Qualitative data are not as widely used as quantitative data because many numerical techniques do not apply to the qualitative data. For example, it does not make sense to find an average hair color or blood type. 

Quantitative data Quantitative data are always numbers and are usually the data of choice because there are many methods available for analyzing the data. Quantitative data are the result of counting or measuring attributes of a population. Amount of money, pulse rate, weight, number of people living in your town, and the number of students who take statistics are examples of quantitative data. Quantitative data may be either discrete or continuous. 

All data that are the result of counting are called quantitative discrete data. These data take on only certain numerical values. If you count the number of phone calls you receive for each day of the week, you might get 0, 1, 2, 3, etc. 

Example 2: Data Sample of Quantitative Continuous Data 

The data are the weights of the backpacks with the books in it. You sample the same five students. The weights (in pounds) of their backpacks are 6.2, 7, 6.8, 9.1, 4. 3. Notice that backpacks carrying three books can have different weights. Weights are quantitative continuous data because weights are measured. 

3. In a management class of 100 student’s three languages are offered as an additional subject viz. Hindi, English and Kannada. There are 28 students taking Hindi, 26 taking Hindi and 16 taking English. There are 12 students taking both Hindi and English, 4 taking Hindi and English and 6 that are taking English and Kannada. In addition, we know that 2 students are taking all the three languages.

If a student is chosen randomly, what is the probability that he/she is not taking any of these three languages?

If a student is chosen randomly, what is the probability that he/ she is taking exactly one language?

Ans. a) Our sample space is all the students in the school.

There are 100 students, so the size of our sample space is 100.

Our event is that a student drawn at random is not taking any language classes. Call this event A

P (A)    = the number of ways A could happen / the size of the sample space

= the number of students taking no language class / 100

So we must find the number of students who are not taking any language class.

Let H be the number of students taking Hindi, E be the number of students taking English, and K be the number of students taking Kannada.

We draw a Venn diagram.

Let students taking Kannada as language be S(K) = 28

Let students taking Hindi as language be S(H) = 28

Let students taking English as language be S(E) = 28

Let students taking Kannada and English be S (K E ) = 12

Let students taking Hindi and English be S ( H E ) = 4

4. List down various measures of central tendency and explain the difference between them?

Ans.

Central tendency

This tutorial uses histograms to illustrate different measures of central tendency. A histogram is a type of graph in which the x-axis lists categories or values for a data set, and the y-axis shows a count of the number of cases falling into each category. For example, if there are 59 men and 48 women in your class, you could represent the information with this histogram:

The categories may be non-numeric, as in the histogram above, or may be numeric, as in the following histogram. The x-axis shows the ages for respondents to a survey and the y-axis reports the frequency or count for occurrences of each age.

From the histogram, can you determine what is the "typical" age of the participants in the survey?   This question could be answered in several different ways, depending on what you really want to know.

Do you want to determine:

The average of the ages? The age which divides the cases into two equal-sized groups -- the "highs" vs. the

"lows"? The most common age?

Questions like these are concerned with determining the central tendency of a group of numbers or data. To answer our question, we want a single number which can somehow represent all of the ages of the people who participated in the survey.

Ways to Measure Central Tendency

The three most commonly-used measures of central tendency are the following.

mean

The sum of the values divided by the number of values--often called the "average."

Add all of the values together. Divide by the number of values to obtain the mean.

Example: The mean of 7, 12, 24, 20, 19 is (7 + 12 + 24 + 20 + 19) / 5 = 16.4.

median

The value which divides the values into two equal halves, with half of the values being lower than the median and half higher than the median.

Sort the values into ascending order. If you have an odd number of values, the median is the middle value. If you have an even number of values, the median is the arithmetic mean (see above) of

the two middle values.

Example: The median of the same five numbers (7, 12, 24, 20, 19) is 19.

mode

The most frequently-occurring value (or values).

Calculate the frequencies for all of the values in the data. The mode is the value (or values) with the highest frequency.

Example: For individuals having the following ages -- 18, 18, 19, 20, 20, 20, 21, and 23, the mode is 20.Check your understanding of these concepts by calculating the mean, median, and mode of the following three sets of numbers.

Which Measure Should You Use?

This histogram shows the distribution of the number of siblings for survey respondents. The mode (i.e., most common number of siblings) is easy to find.  Can you also determine the median simply by inspection?   What about the mean?

You should see two copies of the histogram. The upper histogram allows you to drag the red vertical line to help locate the median. Numbers on either side of the red line show you how many values exist above and below the line.

The lower histogram allows you to move a triangle within the range of the distribution which acts like a fulcrum for a see-saw. The mean is located at the point where the histogram is balanced. Use these tools -- the red vertical line and the fulcrum -- to find the median and mean of the data.

Now write down which of these three measures of central tendency (mean, median, or mode) you think best describes the "typical" number of siblings of the respondents. Explain why you

chose the one you did.

You can use the histogram activity to explore other variables from the the 1993 General Social Survey. The available variables appear under the "Dataset" menu in the histogram window. Look at several of the variables, and use the tools to find  the mean and median for each one.

Notice that not all measures of central tendency are appropriate for all kinds of variables. For example,

For nominal data (such as sex or race), the mode is the only valid measure. For ordinal data (such as salary categories), only the mode and median can be used.

Now explain in your own words how the three measures of central tendency differ from one another. In the space below, briefly answer the following three questions:

1. Why is the mean not appropriate for some types of data?2. When do you want to use the median rather than the mean?3. When would the mode be most appropriate?

5. Define population and sampling unit for selecting a random sample in each of the following cases.

a) Hundred voters from a constituency b) Twenty stocks of National Stock Exchange c) Fifty account holders of State Bank of India d) Twenty employees of Tata motors.

Ans.

Statistical survey or enquiries deal with studying various characteristics of unit belonging to a group. The group consisting of all the units is called Universe or Population Sample is a finite subset of a population. 

A sample is drawn from a population to estimate the characteristics of the population. Sampling is a tool which enables us to draw conclusions about the characteristics of the population.

In sampling there are two types namely discrete and the other is the continuous. Discrete sampling is that the data given are of the finite and their calculations are made easy. Continuous sampling is one where the data are of infinite form. Its intervals are indicated by <, >, greater than but lesser than, lesser than and greater than. 

The finite number of items in a sample is size. In practice samples greater than 30 are large samples and if less it is small samples.

A measure associated with the entire population is called as population parameter or just a parameter.

Given a population, suppose we consider all possible samples of a certain size N that can be drawn from the population.

For each sample supposewe compute a statistic such as mean, standard deviation etc. Thesesample vary from sample to sample. We group these different statistics according to their frequencies which is called as frequency distribution toformed so called as sampling distribution. Standard deviation of asampling distribution is called its standard error.

Suppose we draw all possible samples of a certain size N from a population and find the mean of X bar of each of these samples.

Frequency distribution of these means is called as sampling distribution of  means. 

If the population is infinite, then, be the standard deviation and mean respectively then the standard deviation denoted by is given by

= / sqrt of N

Is used to calculate the standard normal variate for the population where its size is more than 30.

6. What is a confidence interval and why it is useful? What is a confidence level?

Ans.

Under a given hypothesis H the sampling distribution of a statistic S is normal distribution with the mean

A normal distribution with the mean and the standard deviation then 

Z = 

is the standard normal variate associated with S sothat for the distribution of z the mean is zero and the standard deviation is 1. Accordingly for z the Z% confidence level is ( -z c , zc) this means that we can be Z% confident that i f the hypothesis H is t rue than the value of z lie between –zc and zc. This is equivalent saying that when H is t r u e t h e r e i s ( 1 0 0 – Z   )   % c h a n c e t h a t t h e v a l u e   o f z   l i e s o u t s i d e t h e interval (-zc . zc)if we reject a true hypothesis H on the grounds that the value of z lies outside the interval (-z, zc) we would be making a type 1error and the probabil i ty of making this error is (100-Z)% the level of  significance.

Confidence level is very much useful as we can predict any assumptions can be made so that it will not lead us to the wrong way even if it doesn’t be so great. As explained the confidence level is between –zc to z and the peak is at 100% which is the best. In some cases we predict but do not consider it , and sometimes we will not predict but hypothesis need it so this is called as the TYPE 1 errors and TYPE 2 errors. According to the levels of the Z the confidence is assured. In the above the field shaded portion is the critical region.

Assignment Set- 2

1. (a) What are the characteristics of a good measure of central tendency?

(b) What are the uses of averages?

Ans.

a). The characteristics of a good   measure   of central tendency are:

Present mass data in a concise form

The mass data is condensed to make the data readable and to use it for further analysis. 

•Facilitate comparison

It is difficult to compare two different sets of mass data. But we can compare those two after computing the averages of individual data sets.While comparing, the same measure of average should be used. It leads to incorrect conclusions when the mean salary of employees is compared with the median salary of the employees.

•Establish relationship between data sets

The average can be used to draw inferences about the unknown relationships between the data sets. Computing the averages of the data sets is helpful for estimating the average of population.

•Provide basis for decision-making

In many fields, such as business, finance, insurance and other sectors, managers compute the averages and draw useful inferences or conclusions for taking effective decisions.

The following are the requisites of a measure of central tendency:

•It should be simple to calculate and easy to understand•It should be based on all values•It should not be affected by extreme values•It should not be affected by sampling fluctuation•It should be rigidly defined•It should be capable of further algebraic treatment

b) Appropriate Situations for the use of Various Averages

1. Arithmetic mean is used when:

a. In depth study of the variable is neededb. The variable is continuous and additive in naturec. The data are in the interval or ratio scaled. When the distribution is symmetrical

2. Median is used when:

a. The variable is discreteb. There exists abnormal valuesc. The distribution is skewedd. The extreme values are missinge. The characteristics studied are qualitativef. The data are on the ordinal scale

3. Mode is used when:

a. The variable is discreteb. There exists abnormal valuesc. The distribution is skewedd. The extreme values are missinge. The characteristics studied are qualitative

4. Geometric mean is used when:

a. The rate of growth, ratios and percentages are to be studiedb. The variable is of multiplicative nature

5. Harmonic mean is used when:

a. The study is related to speed; timeb. Average of rates which produce equal effects has to be found.

2. Your company has launched a new product .Your company is a reputed company with 50% market share of similar range of products. Your competitors also enter with their new products equivalent to your new product. Based on your earlier experience, you initially estimated that, your market share of the new product would be 50%. You carry out random sampling of 25 customers who have purchased the new product ad realize that only eight of them have actually purchased your product. Plan a hypothesis test to check whether you are likely to have a half of market share.

Ans.

A company has launched a new product. Our earlier experience, initially estimated that, market share of the new product would be 50%.Any hypothesis which specifies the population distribution completely. Statistical hypothesis testing plays a fundamental role.

The usual line of reasoning is as follows:

1. We start with a research hypothesis of which the truth is unknown.

2. The first step is to state the relevant null and alternative hypotheses. This is important a s   i s -s t a t i n g   t h e   h y p o t h e s e s   w i l l   m u d d y   t h e   r e s t   o f   t h e   p r o c e s s .

Specifically, the null hypothesis allows attaching an attribute: it should be chosen in such a way that i t a l lows us to conclude whether the al ternat ive hypothesis can either be accepted or stays undecided as it was before the test.

3. The second step is to  consider the  stat is t ical assumptions  being made about thesample in doing the test; for example, assumptions about the statisticalindependence or about the form of the distr ibutions of the observat ions. This is equally important as invalid assumptions will mean that the results of the test are invalid.

4. Decide which test is appropriate, and stating the relevant test statistic T.

5. Derive the distribution of the test statistic under the null hypothesis from theassumptions. In standard cases this will be a well-known result. For example the test statistics may follow a Student's t distribution or a normal distribution.

6. The distribution of the test statistic partitions the possible values of T into those for which the null-hypothesis is rejected, the so called critical region, and those for which it is not.

7. Compute from the observations the observed value of the test statistic T.

8. Decide to either fail to reject the null hypothesis or reject it in favor of the alternative. The decision rule is to reject the null hypothesis if the observed value is in the critical region, and to accept or "fail to reject" the hypothesis otherwise.

It is important to note the philosophical difference between accepting the null hypothesis and simply failing to reject it. The "fail to reject" terminology highlights the fact that the null hypothesis is assumed to be true from the start of the test; if there is a lack of evidence against it, it simply continues to be assumed true. The phrase "accept the null hypothesis" may suggest it has been proved simply because it has not been disproved, a logical fallacy known as the argument from ignorance. Unless a test with particularly high power is used, the idea of "accepting" the null hypothesis may be dangerous. Nonetheless the terminology is prevalent throughout statistics, where its meaning is well understood. Alternatively, if the test ing procedure  forces us   to  reject   the null   hypothesis  (H-null) ,  we can accept   the alternative hypothesis (H-alt) and we conclude that the research hypothesis is supported by the data. This fact expresses that our procedure is based on probabilistic considerations in the sense we accept that using another set could lead us to a different conclusion.

3. The upper and the lower quartile income of a group of workers are Rs 8 and Rs 3per day respectively. Calculate the Quartile deviations and its coefficient?

Ans.

Quartile   Deviation:

 It is based on the lower quartile and the upper quartile. The difference is called the inter quartile range. The differencedivided by is called semi-inter-quartile range or the quartile deviation. ThusQuartile Deviation (Q.D)

In this question = 3 and = 8Q . D   = 8 - 3 2 = 2 . 5

Here Quartile deviation is Rs 2.5 per day.

Coefficient of Quartile Deviation Coefficient of Quartile Deviation:

A relative measure of dispersion based on the quartile deviation is called the coefficient of quartile deviation. It is defined as Here In this question = 3 and = 8

Coefficient of Quartile Deviation is Rs 0.455 per day

0.455

=

5

11

=

Coefficient of Quartile Deviation

=

2

8 + 3

2

8 – 3.

4. The cost of living index number on a certain data was 200. From the base period, the percentage increases in prices were—Rent Rs 60, clothing Rs 250, Fuel and Light Rs 150 and Miscellaneous Rs 120. The weights for different groups were food 60, Rent 16, clothing 12, Fuel and Light 8 and Miscellaneous 4.

Ans.

Arran ging   the   data   in   tabular form   for e asy   repres entat ion

ITEM P W(Wt) wPRENT 60 16 960

CLOTHING 250 12 3000FUEL & LIGHT 150 8 1200

MISCELLANEOUS 120 4 480FOOD - 60 60

∑ W=100 ∑ wP=5700

P 01= ∑wP ∑ W

= 5700100 = 57

Hence living Index No is 57.

5. Education seems to be a difficult field in which to use quality techniques. One possible outcome measures for colleges is the graduation rate (the percentage of the students matriculating who graduate on time). Would you recommend using P or R charts to examine graduation rates at a school? Would this be a good measure of Quality?

Ans.

In statistical quality control, the p-chart is a type of control chart used to monitor the proportion of nonconforming units in a sample, where the sample proportion non conforming is defined as the ratio of the number of nonconforming units to the sample size, n. The p-chart only accommodates “pass"/"fail"-type inspection as determined by one or more go-no go gauges or tests, effectively applying the specifications to the data before they’re plot ted on the chart . Other types of control charts display the magnitude of the quali ty characteristic under study, making troubleshooting possible directly from those charts. Some practitioners have pointed out that the p-chart is sensitive to the underlyingassumptions, using control limits derived from the binomial distribution rather than from the observed sample variance. Due to this sensitivity to the underlying assumptions, p-charts are often implemented incorrectly, with control limits that are either too wide or too narrow, leading to incorrect decisions regarding process stability. A p-chart is a form of the Individuals chart (also referred to as "XmR" or "ImR"), and these practitioners recommend the individuals chart as a more robust alternative for count-based data.

R Chart:

Range charts are used when you can rationally collect measurements in groups (subgroups) of between two and ten observations. Each subgroup represents a “snapshot" of the process at a given point in time. The charts' x-axes are time based, so that the charts show a history of the process. For this reason, you must have data that is time-ordered; that is, entered in the sequence from which it was generated. If this is not the case, then trends or shifts in the process may not be detected, but instead attributed to random (common cause) variation. For subgroup sizes greater than ten, use X-bar / Sigma charts, since the range statistic is apoor est imator  of  process  sigma for   large subgroups.   In fact ,   the subgroup sigma is ALWAYS a better estimate of subgroup variation than subgroup range. The popularity of the Range chart is only due to its ease of calculation, dating to its use before the advent of computers. For subgroup sizes equal to one, an Individual-X / Moving Range chart can be used, as well as EWMA or Cu Sum charts. X-bar Charts are efficient at detecting relatively large shifts in the process average, typically shifts of +-1.5 sigma or larger. The larger the subgroup, the more sensitive the chart will be to shifts, providing a Rational Subgroup can be formed.

Hence, R Chart will be a good measure of quality instead of P chart.

6. (a) Why do we use a chi-square test?

(b) Why do we use analysis of variance?

Ans.

Chi-square test

Chi-Square test is a non-parametric test . I t is used to test the independence of  attributes, goodness of fit and specified variance. The Chi-Square test does not require any assumptions regarding the shape of the population distribution from which the sample was drawn. Chi-Square test assumes that samples are drawn at random and external forces, if any, act on them in equal magnitude. Chi-Square distribution is a family of distributions. For every degree of freedom, there will be one chi-square distribution. An important criterion for applying the Chi-Square test is that the sample size should be very large. None of the theoretical expected values calculated should be less than five. The important applications of Chi-Square test are the tests for independence of attributes, the test of goodness of fit and the test for specified variance.

The chi-square (c2) test measures the alignment between two sets of frequency measures. These must be categorical counts and not percentages or ratios measures (for these, use another correlation test). Note that the frequency numbers should be significant and be atleast above 5 (although an occasional lower figure may be possible, as long as they are not a part of a pattern of low figures).

Goodness of fit: A common use is to assess whether a measured/observed set of   measures follows an expected pattern. The expected frequency may be determined from prior knowledge (such as a previous year's exam results) or by calculation of an average from the given data. The null hypothesis, H0 is that the two sets of measures are not significantly different.

Independence: The chi-square test can be used in the reverse manner to goodness of fit. If the two sets of measures are compared, then just as you can show they align, you can also determine if they do not align. The null hypothesis here is that the two sets of measures are similar.

The main difference in goodness-of-fit vs. independence assessments is in the use of the Chi Square table. For goodness of fit, attention is on 0.05, 0.01 or 0.001 figures. For independence, it is on 0.95 or 0.99 figures (this is why the table has two ends to it).

Analysis of variance

Let's start with the basic concept of a variance. It is simply the difference between what you expected and what you really received. If you expected something to cost $1 and i t , in fact , cost $1.25, then you have a variance of $0.25 more than expected. This , of  course, means that you spent $0.25 more than what you planned. When you are calculating your variances, take materiality into consideration. If you have a variance of $0.25, that isn't a big deal if the quantity produced is very small. However, as the production run increases, then that variance can add up quickly. Most projects generate tons of variances every day. To avoid a t idal wave of numbers that are inconsequential , instead focus on the large variances. For example, it is far more important to find out why there is a $10,000 cost variance than to spend two days determining why an expense report was $75 over budget. We want to do variance analysis in order to learn. One of the easiest and most objective ways to see that things need to change is to watch the financials and ask questions. Don't get me wrong: You cannot and should not base important decisions solely on financial data. You must use the data as a basis to understand areas for further analysis. For example, if a bandsaw is a bott leneck, then go to the department and ask why. The reasons for the variance may range from the normal operator being out sick, to a worn blade, to there not being enough crewing and a great deal of overtime being incurred. Use the numbers to highlight areas to investigate, but do not make decisions without first investigating further.

Point in time variances, meaning singular occurrences, can help some. To make real gains, look at trends over time. If our earlier variance of $0.25 is judged as a one-time event, is that good or bad? We cannot tell with just one value, so let's look at the trend over time. If we see that the negative variance over time was $0.01, $0.05, $0.10, $0.12 and$0.25, then we can see that there apparently is a steady trend of increasing costs and, if large enough to be material, should be investigated. Yes, this can take a lot of time if done manually. However, spreadsheets and computer systems can be used to generate real-time variance reports that are incredibly useful with little to no work to actually run the report.

Variance analysis and cost accounting in general are very interesting fields with a great deal of specialized knowledge. By using variance analysis to identify areas of concern, management has another tool to monitor project and organizational health. People reviewing the variances should focus on the important exceptions so management can become aware of changes in the organization, the environment and so on. Without this information, management risks blindly proceeding down a path that cannot be judged as good or bad.

THANK YOU