Statistics and Data Analysis in Various Fields

The provided document contains a list of research papers, articles, and conference proceedings that showcase the diverse applications of statistics and data analysis across different disciplines. The sources cover topics such as quality engineering, innovation, big data analytics, environmental management, and more. The papers highlight the importance of statistical methods in various fields, from business and operations management to construction engineering and environmental science.

STATISTICS FOR
MANAGEMENT

Table of Contents
INTRODUCTION...........................................................................................................................1
TASK 1............................................................................................................................................1
(a) Investigating Hypothesis under Public Sector.......................................................................1
(b) Investigating Hypothesis under Private Sector......................................................................3
(c) Using excel to generate Earnings-Time Chart for each group..............................................4
(d) Calculation of Annual Growth Rate......................................................................................6
.........................................................................................................................................................7
TASK 2............................................................................................................................................9
(i) Employing Cumulative Histogram in estimation of Median and Quartiles for Leisure
Centre..........................................................................................................................................9
Computation of Additive Frequency...........................................................................................9
Ogive.........................................................................................................................................10
(ii) Computation of Central Tendency and Dispersion:............................................................13
(a) Comparison between the statistical measurements of two leisure centre............................14
TASK 3..........................................................................................................................................15
(a) Computing optimal level of stock .......................................................................................15
(b) Number of Orders to be placed for tee-shirts......................................................................16
(c) Computing Inventory Policy Cost.......................................................................................16
(d) Current Service Level to Customers....................................................................................17
(e) Ascertaining re-ordering levels to fulfil desired service levels...........................................17
TASK 4..........................................................................................................................................17
CONCLUSION..............................................................................................................................20
REFERENCES..............................................................................................................................21

INTRODUCTION
Statistics relates to the tools and techniques used in the process of decision-making in an
organisation. It takes into account important information derived in the form of qualitative and
quantitative data to draw conclusions and take strategic actions by the business. Utilization of
statistics in a business scenario helps in increasing productivity, reducing costs and measuring
uncertainty in a scientific manner (Baglin and Da Costa, 2012).
This report aims at depicting detailed account in regards to the measures of central
tendency, dispersion and variability. Moreover, the report is divided into four tasks which relate
to hypothesis testing, ogive, inventory management model and production of charts through data
collected.
TASK 1
The term 'hypothesis' refers to a statement or explanation proposed in relation to the
happening or non-happening of a particular event. A hypothesis is usually formed on the basis of
past experiences, observations and data collected to infer judgements regarding the given
statement. In the context of Business Statistics, this methodology is adopted in order to
experiment with various assumptions and plans undertaken by the organisation to ascertain how
successful they would be in implementation of such scenarios in the real world (Borio, 2013).
When forming a hypothesis, a researcher usually takes two scenarios for the proposed
explanation. One that is true (H0) and one that is not (H1). Hereafter, the researcher employs
appropriate statistical techniques in order to spot patterns and connect information to ascertain
whether the formed hypothesis is true or not. In case, the hypothesis proves to be true, it is said
that H0 is accepted and H1 is rejected. This process of experimentation is known as 'Investigation
of Hypothesis' (BOUKACEM‐ZEGHMOURI and Schöpfel, 2012).
For the given case scenario, the data has been extracted for public as well as private
sector and has been construed on the basis of sex. The following hypothesis have been
formulated in each scenario to ascertain the truth in such assertions:
(a) Investigating Hypothesis under Public Sector
Under this scenario, the following hypothesis has been formulated:
H0: Earnings of men is not substantially higher to that of women in Public Sector.
H1: Earnings of men is substantially higher to that of women in Public Sector.
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The above calculations indicate that there has been a difference in pay for men and women
constantly for the time-period of 2009 to 2016. The fourth column of the table depicts the
difference between the mean annual gross earnings. The fifth or last column shows the
percentage in gap on a yearly basis. In all the years taken into account, the minimum pay for
Men have been £30,638 in 2009 and for women it has been £25,224 in 2009. This gives a pay-
gap of £5,414 between the two genders. However, this gap has reduced over the years to as low
as £5,173.
One can observe that the highest percentage change in the earnings is recorded in 2010
where the difference reduced by 4.86% rendering a value of £5,151. Another subsequent
decrease in the gap lead to a further decrease resulting in the gap to be as low as £4,910. If the
2009 to 2011 period is evaluated, there has been a total decrease in pay gap by 9.31%. However,
these reductions were recovered with an increase in gap by 4.95% in 2012. After this point,
almost similar difference has been maintained until 2014, which again reduced by 0.58% in
2014. In 2015, the sector has experienced one of the biggest pay-gaps between men and women
mean annual gross earnings with an increment of 11.83%.
If analysed closely, it is determined that while the earnings for men changed by 2.45%,
the earnings for women in this industry rose by 0.70%. This inequivalent increase in the earnings
resulted in broadening of the gap and hence the percentage. Furthermore, 2016 has been a year of
recovery for the Public Sector as they were able to reduce the gap by 8.84% which is remarkable
(Box and Woodall, 2012). Again, the contribution for this recovery goes to increase in income
for men to £34,011 from £33,685 rendering a 0.97% whereas women's earnings from £27,900 to
£28,053 attributing to 0.55% growth from previous year. Between 2009 and 2016, the total
increase in Men's earnings is that of 11% whereas for Women it comes to 11.22%.
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One the basis of these findings, it can be concluded that the earnings of men are not
substantially higher than that of women employed in the Public Sector. This means that H0 is
accepted and H1 is rejected.
(b) Investigating Hypothesis under Private Sector
Under this scenario, the following hypothesis has been formulated:
H0: Earnings of men is not substantially higher to that of women in Private Sector.
H1: Earnings of men is substantially higher to that of women in Private Sector.
The above figure, similar to the previous table, showcases the bifurcation of mean annual
gross earnings in the Private Sector for males and females employed in the industry. Again, the
above table includes a column that depicts the difference of pay between the two genders along
with a year-on-year percentage change in this difference regarding the same. For 2009 to 2016,
the total increment in the earnings of Men has been by £2,047 and that of women has been
observed by £2,700. Hence, the growth in pay to both the sex has been by 7.41% and 13.81%
respectively. This indicates that the income earned by women is growing rapidly in comparison
to men in the given time-period.
Unlike the Public Sector, this sector has experienced lesser frequency of change in gaps
for the period of 2009 to 2016. Again, the highest change in gap was felt in 2010 where the gap
reduced by 7.59%. The main cause for this can be attributed to the decrease in income earned by
men and women which went from £27,632 in 2009 to £27,000 in 2010 accounting for a 2.29%
decline in pay for this demographic. For women the pay declined from £19,551 in 2009 to
£19,532 in 2010. This reported a total decline of 0.10% for the concerned segment. Hereafter,
there has been an increase for both Men and Women till 2016. Both segments have been able to
minimize the pay-gap by £7,425 in 2014 and then to £7,428 in 2016 indicating that proper
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measures have been taken by industry players to control the gap and encourage recognition as
well as equality in the work environment for both genders (George, Haas and Pentland, 2014).
On the basis of such observations, the hypothesis holds true for the data extracted
resulting in acceptance of H0 while H1 is rejected.
(c) Using excel to generate Earnings-Time Chart for each group
 Earnings in Public Sector for Men:
 Earnings in Public Sector for Women:
4

 Earnings in Private Sector for Men:
 Earnings in Public Sector for Women:
5

(d) Calculation of Annual Growth Rate
In Finance, 'Annual Growth Rate' refers to the rate at which an organisation grows,
especially in terms of profits and revenue from one year to another. It is an important tool used in
taking important investment-related decisions by organisation's management. Calculation of
yearly growth rate promotes comparability in the business in order to ascertain business
performance. It also helps in planning the future course of business by undertaking projects and
opportunities critical to the success of the enterprise (Granato, 2014). The following equation is
employed in order to calculate annual growth rate for a particular value of measurement:
Annual Growth Rate (%) = [(Value1- Value0)/(Value0)]*100 where,
Value1 = Measurement in Current Year
Value0 = Measurement in Previous Year
In the context of continuing the study discussed above, annual growth rate for each
demographic has been calculated for both Public as well as Private Sector. These calculations
have been provided below:
 Earnings in Public Sector for Men:
6

The annual growth rate for men has been ranging between 0.37% to 2.45%. For 2016, it
has been reduced to 0.97% from 2.45% in 2015. This means that there has been a state of
stagnancy in the income earned by the male workforce of this industry. The overall growth rate
for this segment has been accounted at 11%.
 Earnings in Public Sector for Women:
The Mean Annual Gross Earnings earned by women in this industry has been highly
uncertain and inconsistent. The figures indicate that there is high volatility pursuant as far as
incomes are concerned for this workforce (Haskin and Krehbiel, 2012). The overall growth rate
for this segment comes to 10.61% which is almost nearer to that of their male counterparts
working in the same conditions. Furthermore, year 2016 has felt the lowest growth at 0.70%
among all the others taken account for this study. This signals a trend that is highly uncertain
prevalent in the industry along with growing recognition towards as well as promotion of gender
equality in the work place in relation to women.
 Earnings in Private Sector for Men:
7

The annual growth rate for Men earnings in the Private Sector increased between 2009 to
2016. The overall growth rate has been 7.41% while the annual growth rates for the demographic
have been depicted above. It can be observed that this sector has been able to increment the
payments made to Men in return of their employment services. This may be due to highly
incentivized systems prevalent in the Private Sector which help in speedy appraisals and increase
in pay for the concerned demographic (Juhola, Ahola and Ahola, 2014).
 Earnings in Private Sector for Women:
The annual growth rate for Women earnings in the Private Sector has been inconsistent
between 2009 to 2016. The overall growth rate has been 13.81% which is much higher than any
other segment of Private as well as Public Sectors combined. On the other hand, year 2016 has
been the highest in terms of growth rate accounted at 3.96% which was last closed by this sector
in 2012 at 3.82%. This indicates that 2016 has been highly favourable for Women demographic
working in Private Sector. Again, this can be attributed to highly incentivized systems prevalent
in the related sector which help in speedy appraisals and increase in pay for the concerned
demographic (Kim, Kumar and Kumar, 2012). Also, increase in productivity as well as
contributions made by them in the workplace.
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TASK 2
(i) Employing Cumulative Histogram in estimation of Median and Quartiles for Leisure Centre
Computation of Additive Frequency
'Additive' or 'Cumulative' Frequency, as the name suggests, is the running total of
frequency for each class in a subsequent manner. This means that in order to find the cumulative
frequencies for the given table, one would add frequency of Class Interval £0 to £10 to the value
of frequency of next interval £10 to £20. This would render the additive value of 27 ( = 4 + 23).
Note that the additive frequency for the last interval would always be equal to the total of
frequency (Kotz and Johnson, N.L. eds., 2012). In the given table, the total of number of leisure
centre staff is same as the the cumulative frequency for hourly earnings falling under £40 to £50
interval, that is, 50.
The above table is also known as Cumulative Frequency Distribution. This distribution
helps in ascertaining the accuracy of data collected as well as likelihood of certain variables to
fall within a certain range of frequency. Apart from this, it aims to provides easy tabulation of
data so as to affirm that all the relevant data has been included during the formulation of the
distribution (Kwon, Lee and Shin, 2014).
The data provided above, shows sample extracted from a survey. This survey was
conducted for 50 employees working for a Leisure Centre situated in London Area. The
examination aimed to find the hourly earnings made at the centre between £0 to £50. In order to
inspire easier understanding of the variables, the earnings have been reorganized in the form of
(hourly earnings) intervals ranging between £0 to £10 and £40 to £50 per hour.
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Ogive
As per the previous discussion, the cumulative frequency helps in affirming how
accurately the relevant data has been included in the sample. This can be plotted in the form of a
curve, called cumulative histogram or Ogive. This graph represents the number of values that are
either below or above a central point in the sample.
The above graph shows a cumulative histogram or ogive derived from the data collected
in the survey for the staff of 50 working in London's Leisure Centre. The two axes here are X
and Y-Axis, where, X- Axis has the upper boundaries plotted on it and Y-Axis includes plotting
of additive frequency for the same.
On closely analysing the Ogive, it can be interpreted that there exists a curvilinear
relationship between the two. As this type of histogram includes running totals, the ogive is
positive as the upper class boundaries go on increasing (Leon, Stewart and Yamagishi, 2012).
There is a steeper growth present in the Ogive until it totals 40 at upper class interval of £20.
This means that most of the values lie between intervals ranging £0 to £20 earnings made per
hour. Here, the values above £40 but under £50 come to 10 (=50-40). Hence, one can know
what is the frequency of each class at every upper class boundary through this graphical
representation of additive frequencies.
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In addition to this, Ogive also helps in ascertaining Median and Quartiles. Median is the
central value of a given data. This measurement of central tendency can be found out in two
ways viz. Using a “Less than Ogive” or “More than Ogive”. These have been presented below:
Hourly Earning (£) No. of Leisure Centre Staff More than Ogive
Cumulative
Frequency
Below 10 4 More than Ogive 0 50
10 but under 20 23 More than 10 46
20 but under 30 13 More than 20 23
30 but under 40 7 More than 30 10
40 but under 50 3 More than 40 3
Total 50
Hourly Earning (£) No. of Leisure Centre Staff Less than Ogive Cumulative Frequency
Below 10 4
Less than Ogive
10 4
10 but under 20 23 Less than 20 27
20 but under 30 13 Less than 30 40
30 but under 40 7 Less than 40 47
40 but under 50 3 Less than 50 50
Total 50
By combining the two tables into one graph, the median value can be ascertained. This
has been showcased below:
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Through this, median comes to point of intersection between Cumulative frequencies of
“More than Ogive” and “Less than Ogive”. Therefore, Median for the sample collected on
Leisure Centre Staff comes to 25.5. This table can also help in computing quartiles using a series
of formulas. Quartiles are the ranges that form the boundaries for the given sample data. A
Quartile can be divided into three kinds based on the level of area covered by them. They are
First (Q1), Second (Q2) and Third Quartile (Q3). Additionally, an interquartile range can also be
computed which is basically the difference between the First and Third Quartiles.
First Quartile or Q1 is the lower half of a sample which includes all those variables that lie
under or are less than the Median value or Second Quartile (Q2). It covers 25% of the total
sample data. On the other hand, the Third Quartile or Q3 is the one that covers 75% of the total
dataset. It aims to ascertain the number of data values that lie above the Second Quartile or Q2
(Otero, Castle and Johnson, 2012). In the context of the given case scenario, these calculations
have been provided below:
First Quartile = (25% of (n+1)) = (0.25* (50+1)) = £12.75
Third Quartile = (75% of (n+1)) = (0.75* (50+1)) = £38.25
Inter-Quartile Range = Q3- Q1 = 38.25 - 12.75 = £25.5
Note that the second quartile is also known as 'Median' which has been calculated above
through Ogive.
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(ii) Computation of Central Tendency and Dispersion:
The measures of Central Tendency and Dispersion includes two main types of concepts
viz. Mean and Standard Deviation. Mean is the average taken for the total of data values present
in a sample set. Whereas Standard Deviation is the measurement of spread that is found among
the data values from a central point. Here, Mean relates to the Central Tendency while Standard
Deviation relates to the extent of dispersion.
The above tabulation is indicative of mean calculations for the Leisure Centre Staff. In
order to calculate mean, firstly, a central point for each class interval has been ascertained. This
is done by averaging the two limits, upper and lower. For the hourly earnings falling above £0
but under £10, the upper limit is the former value while the lower limit is £10. Here, the middle
point comes to £5 which is ascertained by adding the two limits and dividing the total (£0+
£10=£10) by 2. This process is repeated for every interval, hence, rendering values 5 to 45. The
Mean here has been calculated by using the following equation:
Mean = ∑fx / ∑f where,
∑fx = Summation of (frequency * central point of intervals)
∑f = Summation of frequency
For the current scenario, the calculations for mean ensue as under:
Mean = ∑fx / ∑f = 1070/50 = £21.4
This means that £21.4 is the average income earned by the staff employed in the Leisure
Centre situated in London.
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For calculating Standard Deviation, the tabulation provided in the preceding section has
been extended along with the mean calculations. Standard Deviation has been ascertained by
employment of given equation :
Standard deviation = √ (∑fx2 / ∑f) - (∑fx / ∑f)2 where,
∑fx2 = Summation of Squared (frequency * central point of intervals)
∑f = Summation of frequency
(∑fx / ∑f)2 = Squared Mean
Here, the deviation is the square root of variance. Variance is another measure of
dispersion that looks at how distant one data value is from another (Steinberg, 2016). Standard
Deviation in relation to the current premise has been provided below:
Standard deviation = √ (∑Fx2 / ∑F) - (∑Fx / ∑F)2 = √[(28050/50) – (21.4)2 ]= √[561 – 457.96]
= √103.04 = 10.15
This measure indicates that the extent of dispersion among the hourly earnings made by
the staff occurs with a £10.15, give or take.
(a) Comparison between the statistical measurements of two leisure centre
Analysing the results from derived data for two leisure centres located in London and
Manchester respectively, the four main points of comparisons are found viz. Median, Mean,
Standard Deviation and Inter-Quartile Range. The central value or median for the two are £25.5
and £14 respectively. This is indicative of the fact that the hourly earnings of London are higher
which can be due to more wages paid to the staff of London instead of Manchester.
Consequently, the average hourly income earned by the London staff is also higher.
However, when looking at the spread among the data collected from London and
Manchester Survey, there is more consistency in Manchester as it has a lower dispersion present
in comparison to London. Also, the number of data values covered by the inter-quartile range of
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London is higher showing more accurate and useful data as compared to its Manchester
counterpart.
TASK 3
Inventory Management System is a computerised framework that aims at trailing stock
levels, orders, gross revenue as well as deliveries. This system helps in keeping a check on the
level of stock sold, stored as well as still in process in the form of 'work-in-progress'. Also, it
gives a detailed description in regards to the amount of stock that is to be bought or renewed in
order to ensure smooth procession of production (Tran, Lester, and Sobin, 2014). Usually, this
framework is utilized by manufacturing enterprises and purchase houses.
(a) Computing optimal level of stock
EOQ or Economic Order Quantity is an inventory model formulated to ensure that
neither shortage nor excess of stock occurs in the organisation while engaging the production
process of goods.
In the context of given premise, Jenny Jones' requires implementation of a fresh
inventory policy that would enable Ms. Jones to sell her popular product line of tee-shirts' 95%
of the time a customer asked for it. The following basic information can be narrated in a tabular
format:
There are two main costs which are scrutinized by the business working on EOQ model. These
are delivery costs and holding costs. Delivery Costs are those that are incurred while goods are
in-transit. This means that when a business makes an purchase requisition order with its supplier
for raw materials, the cost incurred in transferring such orders to the enterprises' grounds from
the storage house of the supplier is known as Delivery Cost (Winter, 2012).
Holding Costs, on the contrary, are those which are borne by a manufacturing enterprise
for storing or keeping the stock in workable conditions. By workable one means to keep in a
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condition that is favourable in maintaining the quality of the goods until processed and sold.
Delivery and Holding Costs have an inverse relationship with each other. When one increases,
the other tends to decline. Therefore, it is important to know at what point these costs would
render to be minimal and also be able to suffice the production needs of the entity. Hence, the
adequate level of stock in the shop can be found by using the following method:
Economic Order Quantity = √ [(2*Annual Demand* Delivery Cost)/ Holding Cost]
= √ [(2*1500 tee-shirts*£5)/£2]
= 86.6 or 87 tee-shirts
Therefore, if Ms. Jenny wants to ensure that no extra cost is incurred by her she needs to
keep a level of 87 tee-shirts at all times.
(b) Number of Orders to be placed for tee-shirts
In order to maintain the optimal level of stock in the shop, Ms. Jenny needs to place the
following amount of orders with her suppliers on a yearly basis:
Number of Orders required to be placed = [(Annual Demand / Economic order Quantity)]
= 1500 tee-shirts/ 87 = 17.24 or 17 orders.
(c) Computing Inventory Policy Cost
Inventory Policy Cost or Total Cost is the expenditure incurred by a business entity in
regards to ordering and holding cost on the economic level of quantity for a given financial
period (Zupic and Čater, 2015). As the owner wants to ensure 95% of successful deliveries to her
customers, she will require to maintain a cost policy in regards to optimal stock level so that she
does not lose her loyal customers to competitive stores working in the shopping centre. For this
purpose, she needs to find out the annual holding and delivery costs. These have been provided
below:
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Hence, she will need to incur a total of £173.21 annually so as to implement the new
policy in a prosperous manner.
(d) Current Service Level to Customers
Current Level of service = Weekly Demand * Availability of tee-shirt
= 30*95%
= 28.5 units
(e) Ascertaining re-ordering levels to fulfil desired service levels
Re-order level (ROQ) = (Lead time*daily average usage)+safety stock
= (28*2)+150
= 206 units
TASK 4
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18

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CONCLUSION
The above report provides a summary of different methodologies employed by business
entities of varying size and nature in regards to managing their operations. The report also
showcases the manner in which different business data is derived from published sources for
generation of substantive information. This included ascertainment of deviations and economic
quantity models. These conclude that in order to minimize costs related to inventory, the
organisations can employ inventory management systems such Economic Order Quantity Model.
Also, changes in indices such as CPI and RPI can be easily seen through charts which prove to
be useful in summarizing important figures for management in a concise manner.
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