How can I read logarithmic scale?

Question

I've got this histograms: How can I read that logarithmic scale? For example, on the histogram 1 there is approximately $10^{-3}$ value at y-axis at 2 value at x-axis. Does it meant that there is a $10^{-3}$ (i.e. 0.001%) chance that there will be such particle at such momentum?

Answer 1

You've asked two questions:

1. How to read a semi-log plot
2. How to interpret a histogram

Your plots are on a semi-log scale, rather than a log-log scale, because the horizontal axis is still linear.

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To interpret the vertical axis, simply count the minor tick marks. Labelling an excerpt of the vertical axis explicitly…

         ┋
    10⁻³ ┣━━━
9 × 10⁻⁴ ┣━
8 × 10⁻⁴ ┣━
7 × 10⁻⁴ ┣━
6 × 10⁻⁴ ┣━
         ┃
5 × 10⁻⁴ ┣━
         ┃
         ┃
4 × 10⁻⁴ ┣━
         ┃
         ┃
         ┃
3 × 10⁻⁴ ┣━
         ┃
         ┃
         ┃
         ┃
2 × 10⁻⁴ ┣━
         ┃
         ┃
         ┃
         ┃
         ┃
         ┃
         ┃
    10⁻⁴ ┣━━━
         ┋

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Histograms normally have integral counts on the vertical axis, but this one has small fractions instead. I assume that the vertical axis has been normalized such that the total number of samples is 1. Therefore, the vertical axis should be interpreted as relative probability density rather than a count.

Histograms are normally bar graphs rather than line plots. Imagine that each + symbol represents a vertical rectangle with its top at the + mark and its sides being halfway between it and its neighbours.

It looks like the + symbols are spaced horizontally such that there are 10 data points per major tick, i.e. at intervals of $0.2 \frac{\mathrm{GeV}}{c}$. Therefore, each data point represents data samples at the marked momentum, $\pm\ 0.1 \frac{\mathrm{GeV}}{c}$.

Taking your example, reading the first histogram at $x = 2$, I would interpret a + symbol at $x = 2.0 \frac{\mathrm{GeV}}{c}$, $y = 6 \times 10^{-4}$ as

$6 \times 10^{-4}$ probability of a collision when the momentum is in the range $2.0 \pm 0.1 \frac{\mathrm{GeV}}{c}$.