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Data Table Statistics

Using and interpreting statistics in the retail sector: a report prepared for departmental managers at JW Stour

• Introduction

Many people in managerial positions have a fear of maths and statistics but there is really no need to be afraid. Statistics when carefully presented are the perfect tools for managers to highlight and explain key management issues.

In this report I will outline a range of tools that you can use in your day-to-day work with JW Stour. All concepts will be clearly explained, a selection of different applications will be highlighted, and a fully worked example will be provided in each section of this report to illuminate how the tool can be applied to the organisation.

• Presenting data using tabulations

What is data? Data is raw and unorganised facts and figures. We present data using tabulations to ensure that the data is more clearly understood. There are different ways of tabulating data and we will explore some of these below.

Frequency tables are useful when we want to show clusters around a central value and they are also useful in showing the degree of difference between observations.

Frequency distributions are an easy way to organise raw data, as shown in the following example:

Suppose we want to display some data relating to sickness absence at JW Stour. The items of data that we have are: 5, 2,2, 3, 4, 4, 0, 2, 3, 23, 1, 0, 3, 2, 5, 1, 3, 1, 5, 5, 0, 4, 2, 0, 4, 4, 4, 5, 5, 5.

The frequencies of occurrence of the different absences can be summarized as:

The table enables us to see more clearly the frequencies of absence, which in turn enables us to make comparisons with other organizations with a relative frequency distribution table or to look at how many employees have absences more than or less than a certain amount with a cumulative frequency distribution.

• Presenting data using diagrammatic techniques

As well as presenting information in tables we can also use diagrams to made the information more visually attractive and more accessible to a wide range of people.

Many of us will remember bar diagrams and pie charts from school. Bar diagrams are ideal to present sales figures, and pie charts are an attractive way to present information about market share but you can choose either diagram to present the information according to your preference. In our example we will look at employee numbers in the different shop floor departments at JW Stour:

Radar diagrams are also useful for providing an attractive way of showing performance against targets, which is particularly applicable to JW Stour’s sales and human resource management functions (managing performance, monitoring sickness absence etc).

In this example we can look at the table to see the sales performance of individual sales staff in the electrical department of JW Stour and compare their performance to their individual targets, and then see the same information in a radar diagram:

• Measures of location and dispersion

We will now look at measures of location or average for a collection of data: the mean, the median, and the mode.

• The mean (or avaerge) is obtained by adding up the individual items of data and dividing the result by the number of items.

• The mode is the most commonly occurring value or item of data or, in other words, the one that occurs most frequently

• The median is the centre or middle item and is identified by placing the data in order and locating the middle item

Measures of location can be misleading unless we also know how dispersed or variable the data is. The range is a simple measure of dispersal and this is identified simply by looking at the minimum and maximum values, however this doesn’t tell us anything about the data within the range.

Another measure of dispersion is the standard deviation. In this method the variance is calculated by taking the mean of the squared differences between each item and the mean value; the square root of this is the standard deviation. By comparing the standard deviations of waiting times at, say the store restaurant and a competitor’s restaurant, we can identify whether customers on average wait longer for service at one rather than the other.

• The application of probability, sampling, estimation and inference techniques

Now we will look at what probability, or identifying the likelihood of events, can do for us as managers. Wouldn’t it be useful to have a sense of the probability that a customer will spend more than, say, £500? We can in fact calculate this:

If the mean expenditure is calculated as £500 we immediately know that the probability of a customer spending £500 is 50/50. If we wanted to calculate the likelihood that a shopper would spend £800 or more we can calculate this by knowing the mean, calculating the standard deviation (which in this case is £150), and having access to statistical tables to get a standardized value we find that the probability that a customer will spend more than £800 is P(x > £80)  =  0.5 - 0.4772   =  0.0228  or 2.28%

Sampling is a useful technique to give us a snapshot of the views, opinions or habits of the wider population.

Random sampling is where each member of the sample has an equal chance of being selected (as in being picked from a hat). Systematic sampling is useful for large sample populations, where say in a sample of 1,000 you pick every twentieth person to give a sample of 50 people. Stratified sampling is similar to systematic sampling but allows you to stratify people according to demographics. Quota sampling is where you determine beforehand that you want a certain sample from a certain type of person (50 men and 50 women for example), and convenience sampling is just seeking participants on an availability basis.

Estimation techniques such as point estimates are useful techniques when we need to get some information about aspects of the population. Suggested uses for estimation techniques include the average age of employees, the proportion of shoppers aged under 30, the variation in age of shoppers measured by the standard deviation. To achieve this we would get a sample and calculate the sample mean, proportions, and standard deviation and these in turn would give us point estimates of the population mean, the population proportion and the population standard deviation.

• The application of hypothesis tests

Hypothesis testing is all about testing whether a prediction about one of more events on people’s behaviour is well founded. A one- way prediction, or one-tailed hypotheses, is as the name implies – one that makes a prediction in a particular direction.

For this example, we will look at the viability of the introduction of a store account facility at JW Stour. One of the finance team members has determined that a new store account facility will be cost-effective only if the mean monthly spending on each customer’s account is more than £170. In a trial of the account facility in the menswear depertment, a random sample of 400 accounts is drawn, for which the sample mean is £178. The accounts are approximately normally distributed with a standard deviation of £65.

Mean to be tested μ=170

Sample Mean 178

Standard deviation =65

Sample size n=400

Significance level= 0.05

Step 1:

State the null and alternative hypotheses:

H₀: μ = 170 à Null hypothesis

Ha: μ >170 à Alternative hypothesis

Step 2:

State the significance level:

α =0.05

Step 3:

Compute the critical value:

This is one-tail (right) test.

From or statistical tables: the critical value of z= 1.645

Step 4:

Compute the test statistic:

Standard deviation of sample means is

Step 5:

Make a decision:

Since the value of the test statistic of z=2.365 is greater than the critical value of z=1.645, it falls in the rejection region, and we can therefore reject the null hypothesis. We therefore conclude that there is statistically significant evidence that a new customer account facility will be cost-effective only if the mean monthly account is more than £170.

In addition to one-way predictions, there are also two-way predictions, called two-tailed hypotheses, that predict differences in two directions.

When we are creating our predictions, we have to be clear whether we are creating a one- or two-way prediction as we treat each slightly differently at the analysis stage. A one-tailed hypothesis has a specific percentage that the difference might occur randomly, while with a two-tailed hypothesis there is double the probability that such differences might occur randomly.

Example of a two-way prediction:

JW Stour has menswear departments at the store’s original location in Stowmarket as well as in the new Ipswich retail complex. The menswear department’s managers have noticed that products that sell well in one store do not always sell well in the other. The manager believes this situation may be attributable to differences in customer demographics, namely the mean age of the customers, at the two locations.

If we assume that the customer age data collected from the two independent simple random samples of JW Stour customers provide the following results:

Sample size n₁ =36

Standard deviation s₁=9

Mean =40

Sample size n₂=49

Standard deviation s₂=10

Mean =35

Significance level = 0.05

Step 1:

State the null and alternative hypotheses:

H₀: μ₁ - μ₂ = 0

Ha: μ₁ - μ₂ ≠ 0

Step 2:

State the significant level:

α = 0.05

Step 3:

Compute the critical value:

For a total α = 0.05, we place a probability of 0.025 on each side.

(= total α /2)

With α/2 = 0.025

The critical z-score =±1.960

This is a two-tail test.

Step 4:

Compute the test statistic:

The observed difference of and is 5

Standard deviation of sample means is σ_x1-x2

= √(σ1^2/n1+σ2^2/n2) ≈ √(S1^2/n1+S2^2/n2) = 2.0714

Test statistics=

Step 5:

Make a decision:

Probability for test statistic =0.992

P-value = 0.016< 0.05

Therefore the first expression of our test Ho, that there was no difference in mean age, can be rejected.

Our conclusion is therefore that there is statistically significant evidence of difference in the mean ages of the people who shop in the Stowmarket menswear department compared to the Ipswich retail unit.

A further method of hypothesis testing is to use the Chi squared test:

We might for instance predict that a higher proportion of female shoppers at JW Stour have regular shopping habits than would male shoppers. The chi square test would help us compare the observed frequencies with the actual numbers of shoppers who fall into each group. The observed frequencies we would know from market research but we would have to estimate the expected frequencies. If the observed results were random then they should closely resemble the expected frequencies, but if the observed frequencies differed significantly from the expected frequencies then the null hypothesis could be supported.

• The application of regression analysis including the use of correlation coefficients

In regression analysis we are looking for a relationship between two variables, the dependent variable (the one being predicted) and the independent variable (which is what is being used to make the prediction).

In the following example of simple regression, the data shows the sales (thousands) and the advertising expenditures (thousands of pounds) for seven major brands of men’s fashion clothing sold in the store:

Let

X (Advertising Expenditures ($)) is independent variables

Y (Sales) is dependent variable.

Using excel, we conduct a regression analysis on the data such that we can determine that there is positive relationship between the two variables.

The simple regression equation is Y hat =14.42X -15.36

The slope b₁ was computed as +14.42. This means that for each increase of 1 unit in X, the average value of Y is estimated to increase by 14.42.

The Y intercept b₀ was computed to be -15.36. The Y intercept represents the average value of Y when X equals 0.

The coefficient of determination r² is 0.96.

Therefore, 96% of the variation in sales can be explained by the variability in advertising expenditures.

We can also utilise multiple regression:

In this example, JW Stour’s in-store restaurant would like to estimate weekly gross revenue as a function of adverting expenditures. Historical data for a sample of 8 weeks follows:

Let

X₁ (Television Advertising (£1000)) is independent variable

X₂ (newspaper advertising (£1000) is independent variable

Y (Weekly Gross Revenue (£1000) is dependent variable

Using excel for the regression analysis, we get the following output:

The multiple regression equation is Y hat =2.29X₁+1.30X₂+83.23

The slope b₁ is computed as +2.29 which means that, for a given amount of weekly adverts on television, the expected revenue of the store restaurant is estimated to increase by 2.29 per weekly for each 1 unit increase of television adverts.

The slope b₂ is computed as +1.30 which means that, for a given amount of weekly ads in newspapers, the anticipated revenue of the store restaurant is estimated to increase by 1.30 per weekly for each 1 unit increase in the newspaper adverts.

These estimates allow departmental managers to predict the likely effect that advertising will have on sales.

The Y intercept b₀ computed as 83.23, estimates the expected amount of restaurant revenue in a week if there is no money spent on television advertising and no money spent on newspaper advertising.

The multiple coefficient of determination R² is 0.92

Therefore, 92% of the variation in revenue can be explained by the variation in the Television advertising and newspaper advertising.

• Forecasting and time series

Predicting what might happen in the future of the store is key for any manager. Trend projections can help managers achieve this by using linear regression to identify a straight line trend (even allowing for random fluctuations).

Consider the time series for menswear sales at JW Stour over the past ten years as shown below:

Using excel we determine that there is positive relationship between the two variables, with sales steadily increasing and the market is steadily expanding:

The trend line is T_t =1.10t +20.40 is the expression for the linear trend component for the menswear sales time series.

Forecasting can also be helpful for mangers where there is a trend and a seasonal element.

This example shows the quarterly sales of television sets, radios/mp3 and dvd players sold in the store’s electrical department:

The summary output in excel gives us the slope of the as

T_t=0.21t+4.83

The slope of 4.83 indicates that over the last ten years, JW Stour has experienced an average deseasonalised increase in sales of television sets, radio/mp3 and dvd recorders of 483 per quarter.

Using this information, we can forecast the trend for the first quarter of year 6 (the 17^th quarter):

T= 0.21 + 4.83 x 17 = 85.68

We then look at the seasonal factors for each quarter derived by excel and identify the index for the first quarter:

Applying this seasonal factor for the first quarter we can get the forecast for quarter 17:

Y = 85.68 x 1.01 = 86.54 which means that we estimate sales of 865.4 televisions, radios /mp3 and dvd players in the first quarter of year 7.

In the same way we can forecast sales for other future quarters.

Conclusions

Having examined various statistical techniques we can clearly see their applicability to our daily work at JW Stour. To enable sound business planning in all of our departments we can utilize these techniques to ensure that our plans are based on grounded predictions. From a human resource perspective, I have learned a lot about how statistics can support and enhance my work; I am hopeful that I have been effective in sharing this experience with you.

References

Bahree, M (2007) Plugging a Math Gap. Forbes. New York: Mar 26, 2007. Vol. 179, Iss. 6; p. 186

Bee, F & R Bee (2005) Managing Information and Statistics. 2^nd edition. CIPD Publishing.

Hague, PN (2004) Market Research in Practice: a Guide to the Basics. Kogan Page.

Rabolt NJ and JK Miler (1997) Concepts and cases in retail and merchandise management. Fairchild Books.

Rugg, G (2007) Using Statistics: A Gentle Introduction. Open University Press.

Tepper, BK (2008) Mathematics for Retail Buying. Fairchild Books.

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