pxr687
commented
1 year ago I like the example, one thing I might add though, is a set of plots illustrating the paragraph below with some simple, idealized data (e.g. to visualise the concept before visiting it later in the chapter):
Ranks and quantile positions give an idea whether the measure is high or low compared to the other values, but they do not immediately tell us whether the measure is exceptional or unusual. To do that, we may want to ask whether the measure falls outside the typical range of values — that is, how the measure compares to the distribution of values. One common way of doing this is to re-express the measures (values) as standard scores, where the standard score for a particular value tells you how far the value is from the center of the distribution, in terms of the typical spread of the distribution. Standard values are particularly useful to allow us to compare different types of measures on a standard scale. They translate the units of measurement into standard and comparable units. We will explain this more towards the end of the chapter.
So maybe a couple of graphs, just visually motivating why you would want a standard score, which might be something like:
import numpy as np
import matplotlib.pyplot as plt
import scipy.stats as stats
set_of_scores_1 = np.random.normal(100, 15, 10000)
set_of_scores_2 = np.random.normal(600, 100, 10000)
# show separate distributions
plt.figure()
plt.hist(set_of_scores_1, color = "blue")
plt.axvline(np.sort(set_of_scores_1)[-20], color = 'cyan', label = 'Score of interest \n(Distribution 1)')
plt.plot([np.percentile(set_of_scores_1, 2.5), np.percentile(set_of_scores_1, 97.5)], [0, 0],
color = "silver", linewidth = 10, label = "95% spread of Distribution 1")
plt.xlabel("Some Variable")
plt.legend(bbox_to_anchor = (1,1))
plt.title("Distribution 1")
plt.show()
plt.figure()
plt.hist(set_of_scores_2, color = 'orange')
plt.axvline(np.sort(set_of_scores_2)[40], color = 'red', label = 'Score of interest \n(Distribution 2)')
plt.plot([np.percentile(set_of_scores_2, 2.55), np.percentile(set_of_scores_2, 97.5)], [0, 0],
color = "gold", linewidth = 10, label = "95% spread of Distribution 2")
plt.xlabel("Some Other Variable")
plt.legend(bbox_to_anchor = (1,1))
plt.title("Distribution 2")
plt.show()
# show why they are hard to compare
plt.figure()
plt.hist(set_of_scores_1, color = "blue", label = "Distribution 1")
plt.hist(set_of_scores_2, color = "orange", label = "Distribution 2")
plt.xlabel("Both Variables")
plt.axvline(np.sort(set_of_scores_1)[-20], color = 'cyan', label = 'Score of interest \n(Distribution 1)')
plt.axvline(np.sort(set_of_scores_2)[40], color = 'red', label = 'Score of interest \n(Distribution 2)')
plt.plot([np.percentile(set_of_scores_1, 2.5), np.percentile(set_of_scores_1, 97.5)], [0, 0],
color = "silver", linewidth = 10, label = "95% spread of Distribution 1")
plt.plot([np.percentile(set_of_scores_2, 2.55), np.percentile(set_of_scores_2, 97.5)], [0, 0],
color = "gold", linewidth = 10, label = "95% spread of Distribution 2")
plt.legend(bbox_to_anchor = (1,1))
plt.title("Why scores are hard to compare across these distributions")
plt.show()
# show why standard scores are useful
set_of_scores_1_standard = stats.zscore(set_of_scores_1)
set_of_scores_2_standard = stats.zscore(set_of_scores_2)
plt.figure()
plt.subplot(1, 2, 1)
plt.hist(set_of_scores_1_standard, color = "blue", label = "Distribution 1")
plt.plot([np.percentile(set_of_scores_1_standard, 2.5), np.percentile(set_of_scores_1_standard, 97.5)], [0, 0],
color = "silver", linewidth = 10, label = "95% spread of Distribution 1")
plt.axvline(np.sort(set_of_scores_1_standard)[-20], color = 'cyan', label = 'Score of interest \n(Distribution 1)')
plt.xlabel("Some Variable")
plt.subplot(1, 2, 2)
plt.hist(set_of_scores_2_standard, color = "orange", label = "Distribution 2")
plt.xlabel("Some Other Variable")
plt.axvline(np.sort(set_of_scores_2_standard)[40], color = 'red', label = 'Score of interest \n(Distribution 2)')
plt.plot([np.percentile(set_of_scores_2_standard, 2.55), np.percentile(set_of_scores_2_standard, 97.5)], [0, 0],
color = "gold", linewidth = 10, label = "95% spread of Distribution 2")
plt.title("Why standard scores are useful for comparing across distributions")
plt.hist([], color = "blue", label = "Distribution 1")
plt.plot([], color = 'cyan', label = 'Score of interest \n(Distribution 1)')
plt.plot([],[], color = "silver", linewidth = 10,
label = "95% spread of Distribution 1")
plt.legend(bbox_to_anchor = (1,1))
plt.show()I'm thinking something with hidden code (maybe just pictures of the plots). But I think plots to explain the content of that paragraph might make it easier to digest?
The district income data is nice for introducing the mechanics of the standard score, but I think that paragraph quoted above is getting at the core use of standard scores. I think the rough sequence of graphs above might be satisfying before the topic is formally revisited later in the chapter? E.g. it would show why the standard scores are useful then later show how they are achieved?
matthew-brett
ben-herbst
@stefanv - any thoughts on direction?