A guide to comment on QMBM statistically & Mathematically calculated answers – Dec 2016, ABE UK

1.      Comment on Mean

Single mean – Just say it is the average number and it has been calculated using all observations and thus reflect the center of the distribution.

Two data set means – first say, it is the average number and it has been calculated using all observations and thus reflect the center of the distribution. This must be done for the individual data set. Later, make comparison using words given in the below table based on situation

Examples:

Data set A ; mean	Data set B; Mean	Nature of comment
23	40	Data Set B mean is more than data set A or data set A mean is significantly smaller than data set B mean. There is difference of (40-23) = 27 in their means
39	40	Both data set means are almost same, suggest almost equal performance

2.      Comment on Standard deviation and Variance

standard deviation is one of the measures used to describe the variability of a distribution

Clearly larger values mean that the set of data is more spread out around the mean, while smaller values mean that the set of data is less spread out around the mean. It may be used as a unit to measure the distance between any two observations.

For example, in the distribution of employees' heights, the distance in the distribution between a height of 160 cm and a height of 189 cm is 29 cm. But as the standard deviation is 7.24 cm, we can also say that the distance between the two observations is (29 / 7.24), or just over 4 standard deviations

Make comparisons with two standard deviations – the higher the value more dispersed the data and the greater spread.

Note that the standard deviation is an absolute measure of dispersion and is expressed in the units in which the observations are measured

3.      Comment on Correlation Coefficients

Use the below rating scales to comment on correlation coefficient R,

4.      Comment on regression coefficients a & b

In this case, the mathematical relationship between X and Y is given as . This is similar to the usual equation of a line in Mathematics, that is, , except for the notation. Here, a is the y-intercept of the regression line of Y on X and b is its gradient. a and b are known as the regression coefficients and are constants to be determined so that prediction is possible.

 Y = a +bX,  as y = mx + c

The ‘hat’ over the response variable Y is used to emphasis on the fact that the y-value is an expected or theoretical value, hence, a forecast. Note also that, if we had several independent variables, the relationship would be a multiple regression model.

Interpretation of regression coefficients

The constant a, the value of y when x = 0, represents the constant value, and that is y-intercept, this values has no effect with the values in x, whereas b (the gradient) is the rate of change of the variable x. The greater the b value, the greater the rate of the change.

5.      Comment on time -series forecasted values

If three and less than three years given; you have to say

-          The forecasted values likely to be inaccurate

-          Reasons; (1) less time period to observe, three years not enough (2) cyclical variations are not accounted due to limited number of years (3) other factors (qualitative factors) which influence on given scene is ignored.

If five and more years given; you have to say

-          The forecasted values likely to be accurate

-          Reasons; (1) time period considered is enough to draw conclusions  (2) cyclical variations are expected to observe with in 5 or more years (3) other factors (qualitative factors)  which influence on given scene is ignored but mathematically it can be backed. 

6.      Comment on Chi-squared test values

If the test statistic value of chi-squared is greater than the critical value at any given level of confidence, reject the null hypotheses/ initial assumption and support the alternative hypothesis.

If the test statistic value of chi-squared is less than the critical value at any given level of confidence, accept the null hypotheses/ initial assumption and reject the alternative hypothesis.

Consider , the df (degree of freedom) , and p value from critical chi-squared values table with great care.

7.      Comment on test statistic values (significance testing)

If the test statistic value of specific tests conducted is greater than the critical value at any given level of confidence, reject the null hypotheses/ initial assumption and support the alternative hypothesis.

If the test statistic value of specific tests is less than the critical value at any given level of confidence, accept the null hypotheses/ initial assumption and reject the alternative hypothesis.

Consider, the critical values of z for given significance level, for one-tailed and two-tailed tests.

8.      Comment on Index values

Index 100 = means the base year index and the given year will be considered as the base year

Index 143 means compared to base year there is an increase of 43 index points

Index 80 means compared to base year there is a decrease of 20 index points

If re-based to the year higher than previous base year than the index value of previous base year will be less than 100, and if re-based year was lower than the previous base year, then the previous base year index will be greater than 100.

Consider the fact that weighted indexes give better results than unweighted simple indexes

Consider the formulae for Laspeyri and Paasche indexes and comment on the index value based on the idea of which period more importance is given in the formulae (old period / new period). This can be easily understood if you count, how much 0(o) & 1(n) repeats in the given situation. 

9.      Comment on coefficient of skewness

If a distribution is symmetrical, the mean, mode and the median all occur at the same point, i.e. right in the middle. But in a skew distribution, the mean and the median lie somewhere along the side with the "tail", although the mode is still at the point where the curve is highest. The more skew the distribution, the greater the distance from the mode to the mean and the median, but these two are always in the same order; working outwards from the mode, the median comes first and then the mean

You are expected to use one of these formulae when an examiner asks for the skewness (or coefficient of skewness) of a distribution. When you do the calculation, remember to get the correct sign (- / +) when subtracting the mode or median from the mean and then you will get negative answers for negatively skew distributions, and positive answers for positively skew distributions. The value of the coefficient of skewness is between -3 and +3, although values below -1 and above +1 are rare and indicate very skewed distributions. Examples of variates (variables) with positive skew distributions include size of incomes of a large group of workers, size of households, length of service in an organization, and age of a workforce. Negative skew distributions occur less frequently. One such example is the age at death for the adult population of the UK.

10.   Comment on trend line in time series graph

-          Upward sloping trend line – positive / increasing trend

-          Downward sloping trend line – negative / decreasing trend

-          Horizontal/ vertical parallel trend line – constant trend – neither decreasing / no increasing

11.   Comment on scatter diagrams

Scatter diagrams comment can be done by graph method, if more scatters fall near the line, the strength of the relationship increases, and if the more scatters moves away from the line of best fit the strength decreases. The nature of trend depends on the gradient line of the curve. If the the line of best fit is upward sloping, then it is positive and if it is downward sloping it is negative. But the line of best fit either a horizontal line or vertical line then the rate of change is 0, which means no gradient.

Now, you may give a though to these scatter diagrams;

12.   Comment on CV (Coefficient of Variation)

The coefficient of variation is a relative measure of dispersion, i.e. it is independent of the units in which the standard deviation is expressed. The coefficient of variation is calculated by expressing the standard deviation as a percentage of the mean:

coefficient of variation (CV) = (standard deviation/ mean) * 100, By using the coefficient of variation you can compare dispersions of various distributions, such as heights measured in centimeters with a distribution of weights measured in kilograms.