Introduction to Quantitative Techniques

Introduction to Quantitative Techniques

The steepness of a straight line is generally determined by the calculation of the value of the gradient (Bonadonna, Ernst and Sparks, 2018). This is effectively achieved by joining two points. For effective on how to calculate the gradient of a line through the use of a formula, construction of a right-angled triangle is considered. Below is a simple demonstration on how to go about it.

Figure 1. Illustration on how to calculate the gradient of a line.

From the above diagram, a right-angled triangle is constructed through joining points A, B with coordinates of (X1: Y1) and (X2: Y2) respectively. The line that is formed after joining points A and B is referred to be the hypotenuse.  The vertical and horizontal length of the right-angled triangle forms the basis of calculating the gradient of line A, B. considering points A and B, the vertical length is described to be the resulting differences in values of Y- axis, while the horizontal lengths are described to be the resulting differences in the values of X-axis. Thus, from the above explanation we can easily depict the formula for calculating the gradient of a straight line is;

Gradient (m) = (y2 – y1) ÷ (x2 – x1) or (y1 – y2) ÷ (x1 – x2).

From the above formula, we can clearly point out the four most significant determinant of a gradient of the line. These are; coordinates X1, Y1 and X2, Y2. In order to efficiently calculate the gradient of a line, these four coordinates ought to be effectively determined.

in accordance with the statistics, the measure of central tendency may be referred to as finding out the centre values in a particular distribution of the given data. The data that is not typically sorted into specific groups; it’s referred to be the ungrouped data. The most common measures of central tendency are the mode, median and mean. These are the most popular measures of central tendency that majority of the individuals are conversant with.  Their different formula includes;

The formula for calculating mean is; ^– x= , this simply means that you first add all the provided data to get its total which the summation then proceed to divide the total with the number of the categories of data that has been provided, which is the “n” value.

Formula for calculating median in ungrouped data; median = (n+12)th

The description is, if the provided distribution of data contains observations that when counted adds up to an even number, then the median is precisely assumed to be the average that results from the prevailing two middle digits. Steps to follow when calculating the median are; first arrange the distribution in ascending order, go ahead and figure out the centre values then find their average.

The formula for calculating mode is observing the most repeated value in a distribution of data.

The formula for calculating mode is; Mode = L + (FM-f1)h /2fm-f1-f2

L – is assumed to be the lower limit in a particular modal class

F.M. – is the data frequency

f1 – is the data frequency of the preceding class

f2 – is the frequency of the data from the succeeding class

h – is the actual size of the existing intervals

2.

Under slope analysis, the slope can literally be positive or negative. The existing differences between a positive and a negative slope can easily be illustrated through graphical representation, as shown below.

The positive slope will always tend to have an upward movement originating from the left side heading to the right-hand side of the graph (Aaronson, Bothun, Mould, et., al, 2019). On the other hand, the negative slope will always tend to have a downward movement, originating from the right-hand side to the leftward direction when presented on the graph. In some cases, the negative slope tends to appear to be more superior as compared to the negative slope. In an organization, a good illustration of the positive slope is demonstrated when the level of the output increases as it leads to an increase in levels of sales hence revenues earned. A positive slope is characterized by the feature of having an upward movement when plotted on a graph. On the contrary, a negative slope is characterized by a large absolute negative value which always indicates the steepness of the line. The result is a steep downward movement of the line under observation. Where the positive line exists, it simply indicates the existence of two values that have a positive relationship. This means that when values in Y-axis increase the same incidence occurs on the X-axis and vice versa. Under such circumstances, when the gradient of the line is calculated, it is always more than zero. On the contrary, when it comes to a negative slope, there is an aspect of the existence of the negative relationship between the Y and X axes variables. When such variables are plotted on the graph, there is the existence of a steep downward movement of the line. Literally, this means that an alteration that causes an increase of the variables of Y-axis results to decrease of the variables in the X-axis and vice versa. Through a graphical comparison of positive and negative slopes, it is evident that the positive lines will always run from the bottom of the left side running to the top rightward direction of the graph while the negative slope will always tend to run from the top left side of the graph heading to the bottom right direction. The greatest value of the slope is assumed to be contained in the steepest line. This can easily be depicted from observing the lines without having to calculate their values.

Figure 2. A graph illustrating the positive slope.

Figure 3. A graph is representing a negative slope.

To effectively derive the value of the steepness of a line, two variables from both X and Y axes are figured out then subtracted from each other. Simply divide the existing differences of two variables from the Y-axis with that of the X-axis. This way, you will efficiently arrive at the best solution.

Figure 4.  Graphical illustration on how to derive the slope of a line through the use of the calculation method.

The above diagram clearly identifies the formulae that should be adopted in order to effectively determine the slope or rather the gradient of a line.

The slope should be calculated as follows:

Slope = (y2-y1) / (x2-x1)

Explanation, the slope is equivalent to the change in Y-axis divided by the change in the X-axis.

3.a.

Figure 5. A table illustrating values that should be considered when determining the mean and the standard deviation.

Explanation of the above table:

1. In the first column, the data of the age of the tourists who visited the area is represented. The varying age groups are clearly displayed, and it’s easy for every individual to note it down.
2. In the second column the number of times that every age group of the tourists who paid a visit to the area is given out. The higher the frequency (f) of a certain age group visited the area illustrates that individuals who lie in such age group are more interested in visiting the area, more so, tend to be active in tourism activities and vice versa. At the end of the column, the total number of times that the tourists from all the groups visited the area is determined.

• The mid-value of every group of the tourist in accordance to their ages that visited the area is determined. The mid-value column is represented by X. The mid-value is comfortably derived through the addition of the upper limit to the lower limit then you divide the resulting totals with two. This process is repeated until the end of the grouped data under consideration.

2. In the fourth column, the values of the third columns are squared. The column is entitled to be X². All the values of X are repeatedly squared respectively, up to the end of the grouped data. On reaching the end, the summation or rather totalling of all the values under the X² column is done.
3. The fifth column is named as f x. Here the values have been effectively determined by multiplication of the values in the second column which named as frequency (f) with the third column named as x, which contains the mid values. At the end of the grouped data, the summation of f x is calculated.
4. The sixth column represents the values of f x^2. This is arrived at through multiplication of the values from the column named as f which is the frequency with the values of the column named as x^2respectively. At the end of the grouped data, the totals of the whole data that is contained in f x^2 column are calculated.

The process of determining the frequency density from a set of grouped data is very easy. The frequency density forms the base of determining the area in a histogram. The values of the frequency-density are correctly on plotted on the Y-axis during the construction of histogram (Delmas and Liu, 2019). To correctly work out the values of the frequency-density, an individual is required to go ahead and divide the values of the frequencies with their corresponding class width in every group of data. To effectively depict the class width, the individual is required to take the higher limit of a particular class of the grouped data and subtract the values of the lower limit in the same group.

Figure 6. A table was portraying the values of age of the tourists in years, the frequency density, the class width and lastly the frequency density.

From the content on the table above we can easily calculate the standard deviation together with the Mean without having to struggle. The formula for calculating the Mean in a grouped data involves dividing the summation of the f x with the prevailing total number of frequencies in a set of given data. The formula for calculating the Mean is illustrated below:

The Mean = ∑fx / ∑f

Hence , ∑fx = 3032.50

In this case ∑f = 100

The Mean; 3032.5÷100=30.33

From the above calculations, we can see that the mean number of tourists who visited the area under consideration is approximated to be 30.33. The explanation is, the number of times the tourist from a particular age group visited the area is approximated to be 30 times in number. For an individual to extract the standard deviation in such kind of a data set, calculation of the mean-variance is the first requirement. After getting the mean-variance, you proceed to find the square root of that value to get the value of the standard deviation. In the calculation of the variance, the values of summation of fx^2 and the overall total of frequency represented by letter ‘N’ becomes a vital requirement.

The arithmetic formula for getting the variance includes:

(σ 2)  = ∑x2/N – (∑X/N)2

∑x = 235

∑x2= (235)2= 55,225

N (total of frequency) = 100

Hence, the variance = (55, 2255÷100) – (235÷100)×2

=552.21 – 5.53 = 546.68

From the above calculation, the value of the variance of the set data is 546.68. The standard deviation will be achieved by finding the square root of the value of the variance, as demonstrated below.

The Standard deviation, (σ)   2/N) – (∑x/N)2}

=

= 22.12

The standard deviation helps in describing how the data is spread when compared to the expected value. Presence of a small value of standard deviation clearly portrays the relative distribution of the values of a set data in a particular distribution. This simply means that the data lies close to the mean value in a group of data hence indicating the presence of a poor distribution of data. On the contrary, the presence of a higher value of standard deviation in a data set as compared to the Mean clearly illustrates the widespread of data in a particular distribution. From the above calculations, our standard deviation is 22.11. Our previous Mean is 30.33. The existing difference between the mean and the standard deviation is 8.21. The difference in value seems to be high, thus an indication that the set data is well spread out. In relation to our set data, the difference between the mean and the standard deviation illustrates that the population of the tourists grouped respectively to their ages who visited the area is well distributed.

In the summary form, we can say that the calculation of the value of the Mean in a data set is very crucial as it indicates the expected number of performance.  Taking a step to calculate the mean values, help in clearing the doubt of the possibility of occurrence of error hence, satisfactory results. The Mean tends to produce a very minimal possibility of an error when compared to all the other values in a data set (Stephens, Smith and Donnelly, 2018). Having been able to evaluate the value of the Mean from a particular distribution, an individual is able to understand complex statistics that would have been difficult for them. The evaluation of the value of the mean range from discrete to continuous data without any struggle. As in the case of categorical data evaluation of the mean value becomes more complicated as the results would result in numerous errors. Skewed and also outliers distributors are the major factors that influence the calculation of the value of the Mean.

When it comes to the evaluation of standard deviation, it is termed to be crucial for incase of proper understanding of the complex data set. People normally tend not to recognize the significance of calculating standard deviation in a distribution hence ends up not giving it the proper attention that it deserves. The step of calculating the mean and the median in a set of data is always done by many individuals, but the most significant move of determining how well the data is distributed in a data set is never taken seriously. Hence, calculating the standard deviation values is basically more significant as it entails finding out or rather measuring how well the data is spread out in a particular distribution. The usage of standard deviation values in a firm is very significant as it helps in a given direction to the formation of productive decisions in the firm through the management by forming the basis of critically analyzing the business performance in the industry. Standard deviation helps in the effective comparison between various types of data sets. Such details are very important mostly to the businesses that operate under much pressure of a competitive market. Standard deviation is also associated with the aspect of portraying the strengths and the weaknesses of the business; hence the management is able to figure out where the improvement is mostly required.

3.b.

Figure 7. A diagram illustrating the construction of a histogram.

During the construction of a histogram, the frequency values are plotted on the Y-axis while the frequency density distribution values are plotted on the X-axis. The varying width of the data bars in the histogram acts a clear indication of how the data is distributed (Marshall, Hobson, Gull, et., al 2017). From the above histogram, the X-axis values indicate the distribution of the tourists in accordance with their age groups in various classes while the Y-axis represents the frequency of data distribution in a particular class of tourists. Complex data are easily represented in a histogram, and they are easily read and understood without struggles. Identification of the distribution of data through observation of the width of the histogram’s bars has been made simple and easy for everyone. Histograms are majorly used in circumstances where there is a continuous distribution of data.

When determining the mode of data in a grouped data, the use of histograms is mostly preferred. The bar that has the highest height is always considered to be the one that comprises the mode in a particular distribution of the data set. The mode is recognized to be the value that appears more frequently in a particular distribution. The mean, median and mode tend to be the same position most probably the central position in case of an asymmetric histogram (Davies, 2020). The steps to be followed when constructing a histogram that will determine the mode in a data set include:

• Plot the data correctly on the histogram graph, where the values in the Y-axis should be of the frequency while those on the X-axis should be of frequency density.
• Frequencies of every class in the grouped data should be plotted correctly. They form the basis of occurrence of different heights of the histogram bars. Proceed and identify the highest bar in the histogram.
• Draw lines that cross each other from the top corner of the highest triangle and join them with the top corners of the adjacent triangles. The joining point is taken to be the mode of that data set. Draw a line perpendicular to the X-axis and read the prevailing values.

In our case, the model is approximated to be 4.1. This is arrived at through reading the values on the histogram in the X-axis.

Conclusion

In conclusion, we can effectively determine the gradient of the line through the use of the formula of change in Y values divided by the change in X values. This helps in determining the steepness of the line. The most significant four determinants of the gradient include Y1, Y2, X1 and X2, which should be accurately determined for the achievement of accurate results. The gradient is associated with the result of negative and positive curves. The steepness of a slope is always determined by the upward tilt that results. The most common central of tendency is mostly used to determine the centre of the data set or rather the expected value. Further determination of the standard deviation of the distribution helps in figuring out how the data is spread out in a particular data set. The standard deviation plays a crucial role in ensuring there is a proper analysis of the business operations of a company; hence productive decisions are made.

References

Aaronson, M., Bothun, G., Mould, J., Huchra, J., Schommer, R.A. and Cornell, M.E., 2019. A distance scale from the infrared magnitude/H.I. velocity-width relations. V-Distance moduli to 10 galaxy clusters, and positive detection of bulk supercluster motion toward the microwave anisotropy. The Astrophysical Journal, 302, pp.536-563.

Bonadonna, C., Ernst, G.G.J. and Sparks, R.S.J., 2018. Thickness variations and volume estimates of tephra fall deposits: the importance of particle Reynolds number. Journal of Volcanology and Geothermal Research, 81(3-4), pp.173-187.

Davies, E.R., 2020. On the noise suppression and image enhancement characteristics of the median, truncated median and mode filters: Pattern Recognition Letters, 7(2), pp.87-97.

Delmas, R. and Liu, Y., 2019. Exploring students’ conceptions of the standard deviation. Statistics Education Research Journal, 4(1), pp.55-82.

Marshall, P.J., Hobson, M.P., Gull, S.F. and Bridle, S.L., 2017. Maximum-entropy weak lens reconstruction: improved methods and application to data. Monthly Notices of the Royal Astronomical Society, 335(4), pp.1037-1048.

Stephens, M., Smith, N.J. and Donnelly, P., 2018. A new statistical method for haplotype reconstruction from population data. The American Journal of Human Genetics, 68(4), pp.978-989.