This essay is about making decisions with sparse data. It uses an example of the batteries in a remote-reading thermometer. The data are “sparse” because each battery lasts a little less than a year, and it is difficult to gather large numbers of cases to analyze. The problem of sparse data is central to the purpose of statistical analysis, and was the inspiration for William Sealy Gosset’s analyses resulting in the t distribution.

Everything runs on batteries, these days. Things that once required a spring, AC power, gasoline, even nothing at all, now require a battery. Clocks and thermometers once just used springs or weights, while thermometers relied on the expansion of mercury or alcohol. Power saws once used only AC power, but now many use batteries. Even lawn mowers and cars are being changed from gasoline power to batteries. According to some sources, consumers purchase as many as three billion dry cell batteries and 93 million lead acid batteries each year. While each battery costs only pennies, consider what would happen if the typical battery life changed from 317 days, to 219 days, using some of the findings considered below. With a change like that, and no change in pricing, the battery companies might earn more, but consumers would notice a reduction in quality. Consumer perception might be something like you may have noticed with incandescent bulbs after the introduction of LED bulbs. Within the time period of a few months, the typical lifespan of incandescent bulbs declined from years, to weeks. Once the change is noticed, consumers can easily feel cheated by unscrupulous manufacturers. I did not anticipate that change in incandescent bulbs, so did not collect data on it. But I did collect data on batteries for the last 11 years.

When we look at the chemistry and electronics of batteries, we know that the length of time it takes for a battery to discharge, depends on the chemistry inside and the load to which it is connected. However, there is another consideration that is only rarely mentioned; the natural variability of the battery’s life. For example, using the same load, two batteries of the same brand and design, might have two different lifetimes. This perspective can be explored with the application of statistics.

Lets use the battery example to see how some decisions might be made with only sparse data, and how an understanding of the normal distribution can inform those decisions.

When we use even the most basic statistics, we can reveal things that otherwise would remain obscure; especially for sparse data. For example, everyone uses the “average” statistic. That’s good. Things that can be described with numbers will have averages. Some averages are greater than others. That can be revealing, and even more revealing is the “average” considered along with the standard deviation. An average plus or minus one standard deviation, acknowledges that not all measured amounts fall at the average. For normally distributed data, 68% of the measurements can be expected to fall within plus or minus one standard deviation. Similarly, 95 % of measured amounts will likely fall within the range of plus or minus two standard deviations, and 99.7% can be expected to fall within plus or minus three standard deviations. Therefore, if we look at the standard deviation, in addition to the average, we understand even more about an average. If we compare the distribution of some measurement to a normal distribution curve that has the same mean and standard deviation. My previous posts on this topic described distributions of blood pressure, pulse rate, and download speeds. Beyond that, normal distributions can describe the length of a cut board, or the gas consumption of a vehicle, and the behavior of sub-atomic particles. The list is endless.

As with every other application of statistics, a study of battery life requires that either you have a whole bunch of batteries operating with the same load, or you have one load and a number of batteries. For this study, I collected battery life data using 24 batteries used in the same device over an 11-year period.

Over the eleven-year period, two size AA lithium batteries of the same brand were used to power a Radio Shack remote sensing thermometer. The batteries were changed when the remote sensor or the home base failed to report a temperature reading. Twelve pairs of batteries were used in all, and each pair was used for a single measurement. The battery lives ranged from 114 days to 515 days, with an average of 317 days, a median of 339 days, and a standard deviation of 98 days. Granted, there is a small sample involved, and it would be better to test hundreds of batteries, but sometimes, extensive sampling is not possible. In this example, the typical battery lifespan is less than a year, so there have been only 12 pairs of batteries installed and measured across the 11-year period. A record was accumulated in an Excel spreadsheet, and the final calculation of the distribution was made in a C++ program and checked in Excel.

At the heart of the analysis is the red line. That line shows what the distribution of battery lives would be, if the battery life is normally distributed: which is highly likely. If we were to completely fill in the normal curve, in this case, that would require 50 to 100 years, which is impractical. So, this little sample of provides a sampling of what is actually involved, and when we superimpose the graph of a normally distributed sample, we get an overall perspective on how future batteries might behave in this application. So in this case we can see that the range from 114 to 515 is a substantial difference that is also indicated in the large standard deviation of 98 days. Still, it is what it is for the moment, and revisions will be possible as future batteries are replaced and their lifetimes tabulated.

Now, from a management perspective, if we have established the normal distribution of battery lives, and then our batteries start repeatedly coming in on the low end of the curve, then it’s time to either replace the device, or get a different brand of battery. If the battery lives start lasting significantly longer, then we might look at whether the device is operating as it should, of if battery designs have improved lifetimes. If thousands of little batteries are used in a car, the individual and collective lifespans of the batteries can become very important. If only a few of the batteries have short lifespans, then perhaps that’s not a problem. But if all of them have short lifespans, that could be a major problem. It all depends on circumstances that this statistic can inform, and an informed individual can address.

One sidebar observation of this project is the lifespan of the Radio Shack remote-reading thermometer system. I don’t know who actually designed and manufactured the device, but the thermometer continues to operate long after Radio Shack’s financial failure and demise.

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