# [Relation of daily sales] Chi-Square, T-Test or Correlation?

#### kggunes

##### New Member
Hello,
Which to use to test whether Monday sales and sales on the other days of the week are independent from each other.

Case:
A promotion offered on Mondays this year, so I want to "test" if the customers are trading off. The idea is to check the relation of weekdays last year and this year. Which method do you think would fit best?

Thank you for the help in advance!

#### kggunes

##### New Member
Sorry for being unclear earlier..

The price of Product A on Mondays offered for a lower price than the rest of the weekdays in 2012 (35 Weeks). As you can guess, the unit sales of Product A is higher on Mondays compared to the rest of the week and compared to the last year. I guess in theory, under usual circumstances (e.g. without price reduction, stable consumer behavior, etc.) the sold units relation between each day (Mon vs. Tue, Mon vs. Wed, etc.) should be same in 2011 and 2012.

So, I would like to see whether customers are trading off with other days (i.e. if they are not buying the Product A on the other weekdays and wait Mondays to buy it). If this is true than results should be skewed towards Monday, isn't it? And of course, is simple correlation enough to prove this?

Hope that clears a bit..

#### kggunes

##### New Member

in 2011, the price of Product A was same in all weekdays.
in 2012, only Mondays offered for a lower price.

#### hlsmith

##### Less is more. Stay pure. Stay poor.
A good first step might be just running the 2x7 chi-sqaure and looking to see if things appear to change. Is there any seasonality in regards to the purchasing of the product. You may think to compare the same 35 weeks of the two years.

#### kggunes

##### New Member
A good first step might be just running the 2x7 chi-sqaure and looking to see if things appear to change. Is there any seasonality in regards to the purchasing of the product. You may think to compare the same 35 weeks of the two years.
Hello hlsmith,

you meant like this

........MO TU WE TH FR SA SO
2011
2012

This shows me if the years are dependent with days or not, isn't it?

#### hlsmith

##### Less is more. Stay pure. Stay poor.
Correct, this will provid you with frequency table data. It does not matter if you run 2x7 or 7x2, I thought the above presentation would allow you to better see the frequencies per day and year in an interpretable way with the naked eye. A significant p-value will inform you that a difference exists, though it won't tell you which cells are different. Subsequent tests will have to tease that out.

#### victorxstc

##### Pirate
Some things came to my mind.

Running a chi-square between the two years can show you is there a difference between the distributions of sales per week in the years, as hlsmith mentioned. So if there was a difference between the two years, you can say that "Ok my sales were significantly higher in these specific days, in 2012, compared to their counterpart days in 2011." But your aim is to answer this question that "if the Monday buyers are actually those who were buying in other days in 2011, now inclined to buy on Mondays? or if there is actually an increase in the sales and that the increase in sales on Mondays is not compromised by a decrease in the other days?". I really don't quite know how can we sure that those Monday buyers are new buyers added to the total number of customers, or are the same previous buyers shifted to Mondays? Because our control group (2011) does not give us clear information about the individual buyers. All we know is number of sales in weeks, and in days.

But there might be at least one way to figure it out, I think. I think we should compare the average number of sales per weeks in the two years. If there was a significant increase in 2012 compared to 2011, we might infer that if the Monday sale was increased in 2012 (according to the chi-square), it is not compromised by a decrease in sales in other days. For this purpose, you need to run a t-test or its nonparametric alternative to compare average +- standard deviation of sales per week in 2011 (n = 52 weeks) compared to the counterpart value in 2012 (n = 35 weeks).

So, if these two were not different and your test power was not low, and the chi-square showed that the sales pattern is significantly skewed to Mondays in 2012 compared to 2011, there is actually a trade-off (as you called it) and that those are coming to buy on Mondays are likely to be those who previously came to buy in other days in 2011. Otherwise, there might be an increase in the total sale, meaning that you have attracted new customers.

Besides it is good to rule out this sentence: "if the Monday prices were not different (like the way it was in 2011), there were no skewness to Monday sales.". For this purpose, you should first run a chi-square goodness-of-fit test (which differs from the normal chi-square) to compare the sales in 2011 weekdays with an evenly distributed sales pattern (when Sunday sale = Monday sale = other days). If this chi-square shows that despite having a good test power, the sales in 2011 were not significantly different than the expected all-equal-days pattern, it might imply that the skewness to Mondays is related only to the price, not to other additional factors such as being the first day of the week.