![]() ![]() Adding the marker at the top of each column improved the appearance, and when I did that, I was able to lighten the color of the columns to make the markers the focal point while still providing a visual connection to integer bases. I did try this first with columns only, but it didn’t make a strong enough visual impression, mainly because the columns had to be thin to stay clustered near the integer bases. In a sense, we’re treating the bases as categories and plotting columns for each ε \varepsilon. The light vertical lines connect the results to the appropriate base, and by not drawing tick marks on the horizontal axis, we avoid any suggestion that the blue and purple values represent non-integer bases. The change in horizontal position for each value of ε \varepsilon prevents overlap. Because the bases are discrete and our audience knows they take on only integer values, we can shift them horizontally just a bit for each value of ε \varepsilon without fear of confusion. Or something very like jittering, anyway. My fix, to avoid both connecting lines and cover-ups, was to apply a technique that Kieran Healy often uses to good effect: jittering. It is easier to see what’s going on than if he’d just done a scatter plot, but there are still problems, mainly because of the common results for different values of ε \varepsilon and the order of plotting that covers up certain parts of the orange and blue plots. Connecting the lines makes it easier to see which values correspond to the same value of ε. ![]() Generally that’s a good thing to do for functions only defined on integers. I first made this plot using discrete markers rather than connected lines. Cook’s reservation is about using connecting lines. But these are small things, easily fixed. The legend doesn’t use consistent numbers of decimal places. The bases are all integers, but the x-axis tick labels are in multiples of 2.5 units. But to keep my graph consistent with his, I will use the same definition of proportion Cook does.Ĭook uses matplotlib to make most of his graphs, and here we can see that he’s allowing its defaults to give the graph a couple of stylistic shortfalls. That should give us a proportion of 0.444 0.444 instead of 0.400 0.400 for the blue ε = 0.10 \varepsilon = 0.10 plot. For example, in the base 10 case shown above, we have 4 near-integers out of 9 non-trivial possibilities. I don’t have any problem with him excluding that trivial case, but I think he should be consistent and divide by b − 1 b-1 instead of b b. He called these levels ε \varepsilon and produced this graph:Ĭook defines his proportions as the number of near-integers divided by the base, b b, but when he counts the near-integers, he doesn’t include the integers that always occur when n = b n = b. “Close” being a matter of taste, he counted for three levels of closeness: within 0.10 of an integer, within 0.05 of an integer, and within 0.02 of an integer. Or we could look at the base 16 analog “hexadecibels.”įor these analog decibels, which I will hereby refer to as “decibels” (in quotes), the equation for the power ratio is r = b n / b r = b^Ĭook calculated the ratios for a large number of bases and counted how many were close to being integers. Is base 10 unique in this regard? If we were to look at the analogs of decibels in other bases, would we see a higher or lower proportion of near integers? For example, we could look at the base 12 (duodecimal) analog of decibels, or “duodecibels” for short. This relatively large set of near-integers sparked Cook’s curiosity. Here’s a table where I’ve highlighted the four ratios that are pretty close to integers. That’s why you often see examples like “a whisper is 30 decibels” or “a lawnmower is 90 decibels.” Those are 30 and 90 decibels above the threshold of hearing.Īn interesting thing about the decibel scale is that the power ratio gets very close to integer values for certain values of n n. Since the denominator of the ratio is always the same, only the numerator given. That’s because the convention with sound is to express loudness as the ratio of a given sound to one that’s on the threshold of human hearing. If your only experience with decibels is as a measure of loudness, the notion that it’s a ratio may seem odd. ![]()
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