2020-09-22-standardization.md

keywords	title	nb_path	layout
fastai	How does data standardization work?	_notebooks/2020-09-22-standardization.ipynb	notebook

{% raw %}

{% endraw %}

With standardization, we transform the data so its distribution's center will become 0 and its standard deviation will become 1.

{% raw %}

import altair as alt
from vega_datasets import data

</div>

{% endraw %}

For this demo, we use the sp500 price index.

{% raw %}

sp500 = data('sp500')
sp500.head()

</div>

<style scoped> .dataframe tbody tr th:only-of-type { vertical-align: middle; }

.dataframe tbody tr th {
    vertical-align: top;
}

.dataframe thead th {
    text-align: right;
}

</style>

	date	price
0	2000-01-01	1394.46
1	2000-02-01	1366.42
2	2000-03-01	1498.58
3	2000-04-01	1452.43
4	2000-05-01	1420.60

{% endraw %}

{% raw %}

alt.Chart(sp500).mark_line(point=True, strokeWidth=1).encode(
    alt.X('date:T'), alt.Y('price'), alt.Tooltip('price')
).configure_point(size=10)

</div>

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function displayChart(vegaEmbed) {
  vegaEmbed(outputDiv, spec, embedOpt)
    .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));
}

if(typeof define === "function" && define.amd) {
  requirejs.config({paths});
  require(["vega-embed"], displayChart, err => showError(`Error loading script: ${err.message}`));
} else if (typeof vegaEmbed === "function") {
  displayChart(vegaEmbed);
} else {
  loadScript("vega")
    .then(() => loadScript("vega-lite"))
    .then(() => loadScript("vega-embed"))
    .catch(showError)
    .then(() => displayChart(vegaEmbed));
}

})({"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}, "point": {"size": 10}}, "data": {"name": "data-776d9ed0e2795915ad0b63d5871c999b"}, "mark": {"type": "line", "point": true, "strokeWidth": 1}, "encoding": {"tooltip": {"type": "quantitative", "field": "price"}, "x": {"type": "temporal", "field": "date"}, "y": {"type": "quantitative", "field": "price"}}, "$schema": "https://vega.github.io/schema/vega-lite/v4.8.1.json", "datasets": {"data-776d9ed0e2795915ad0b63d5871c999b": [{"date": "2000-01-01T00:00:00", "price": 1394.46}, {"date": "2000-02-01T00:00:00", "price": 1366.42}, {"date": "2000-03-01T00:00:00", "price": 1498.58}, {"date": "2000-04-01T00:00:00", "price": 1452.43}, {"date": "2000-05-01T00:00:00", "price": 1420.6}, {"date": "2000-06-01T00:00:00", "price": 1454.6}, {"date": "2000-07-01T00:00:00", "price": 1430.83}, {"date": "2000-08-01T00:00:00", "price": 1517.68}, {"date": "2000-09-01T00:00:00", "price": 1436.51}, {"date": "2000-10-01T00:00:00", "price": 1429.4}, {"date": "2000-11-01T00:00:00", "price": 1314.95}, {"date": "2000-12-01T00:00:00", "price": 1320.28}, {"date": "2001-01-01T00:00:00", "price": 1366.01}, {"date": "2001-02-01T00:00:00", "price": 1239.94}, {"date": "2001-03-01T00:00:00", "price": 1160.33}, {"date": "2001-04-01T00:00:00", "price": 1249.46}, {"date": "2001-05-01T00:00:00", "price": 1255.82}, {"date": "2001-06-01T00:00:00", "price": 1224.38}, {"date": "2001-07-01T00:00:00", "price": 1211.23}, {"date": "2001-08-01T00:00:00", "price": 1133.58}, {"date": "2001-09-01T00:00:00", "price": 1040.94}, {"date": "2001-10-01T00:00:00", "price": 1059.78}, {"date": "2001-11-01T00:00:00", "price": 1139.45}, {"date": "2001-12-01T00:00:00", "price": 1148.08}, {"date": "2002-01-01T00:00:00", "price": 1130.2}, {"date": "2002-02-01T00:00:00", "price": 1106.73}, {"date": "2002-03-01T00:00:00", "price": 1147.39}, {"date": "2002-04-01T00:00:00", "price": 1076.92}, {"date": "2002-05-01T00:00:00", "price": 1067.14}, {"date": "2002-06-01T00:00:00", "price": 989.82}, {"date": "2002-07-01T00:00:00", "price": 911.62}, {"date": "2002-08-01T00:00:00", "price": 916.07}, {"date": "2002-09-01T00:00:00", "price": 815.28}, {"date": "2002-10-01T00:00:00", "price": 885.76}, {"date": "2002-11-01T00:00:00", "price": 936.31}, {"date": "2002-12-01T00:00:00", "price": 879.82}, {"date": "2003-01-01T00:00:00", "price": 855.7}, {"date": "2003-02-01T00:00:00", "price": 841.15}, {"date": "2003-03-01T00:00:00", "price": 848.18}, {"date": "2003-04-01T00:00:00", "price": 916.92}, {"date": "2003-05-01T00:00:00", "price": 963.59}, {"date": "2003-06-01T00:00:00", "price": 974.5}, {"date": "2003-07-01T00:00:00", "price": 990.31}, {"date": "2003-08-01T00:00:00", "price": 1008.01}, {"date": "2003-09-01T00:00:00", "price": 995.97}, {"date": "2003-10-01T00:00:00", "price": 1050.71}, {"date": "2003-11-01T00:00:00", "price": 1058.2}, {"date": "2003-12-01T00:00:00", "price": 1111.92}, {"date": "2004-01-01T00:00:00", "price": 1131.13}, {"date": "2004-02-01T00:00:00", "price": 1144.94}, {"date": "2004-03-01T00:00:00", "price": 1126.21}, {"date": "2004-04-01T00:00:00", "price": 1107.3}, {"date": "2004-05-01T00:00:00", "price": 1120.68}, {"date": "2004-06-01T00:00:00", "price": 1140.84}, {"date": "2004-07-01T00:00:00", "price": 1101.72}, {"date": "2004-08-01T00:00:00", "price": 1104.24}, {"date": "2004-09-01T00:00:00", "price": 1114.58}, {"date": "2004-10-01T00:00:00", "price": 1130.2}, {"date": "2004-11-01T00:00:00", "price": 1173.82}, {"date": "2004-12-01T00:00:00", "price": 1211.92}, {"date": "2005-01-01T00:00:00", "price": 1181.27}, {"date": "2005-02-01T00:00:00", "price": 1203.6}, {"date": "2005-03-01T00:00:00", "price": 1180.59}, {"date": "2005-04-01T00:00:00", "price": 1156.85}, {"date": "2005-05-01T00:00:00", "price": 1191.5}, {"date": "2005-06-01T00:00:00", "price": 1191.33}, {"date": "2005-07-01T00:00:00", "price": 1234.18}, {"date": "2005-08-01T00:00:00", "price": 1220.33}, {"date": "2005-09-01T00:00:00", "price": 1228.81}, {"date": "2005-10-01T00:00:00", "price": 1207.01}, {"date": "2005-11-01T00:00:00", "price": 1249.48}, {"date": "2005-12-01T00:00:00", "price": 1248.29}, {"date": "2006-01-01T00:00:00", "price": 1280.08}, {"date": "2006-02-01T00:00:00", "price": 1280.66}, {"date": "2006-03-01T00:00:00", "price": 1294.87}, {"date": "2006-04-01T00:00:00", "price": 1310.61}, {"date": "2006-05-01T00:00:00", "price": 1270.09}, {"date": "2006-06-01T00:00:00", "price": 1270.2}, {"date": "2006-07-01T00:00:00", "price": 1276.66}, {"date": "2006-08-01T00:00:00", "price": 1303.82}, {"date": "2006-09-01T00:00:00", "price": 1335.85}, {"date": "2006-10-01T00:00:00", "price": 1377.94}, {"date": "2006-11-01T00:00:00", "price": 1400.63}, {"date": "2006-12-01T00:00:00", "price": 1418.3}, {"date": "2007-01-01T00:00:00", "price": 1438.24}, {"date": "2007-02-01T00:00:00", "price": 1406.82}, {"date": "2007-03-01T00:00:00", "price": 1420.86}, {"date": "2007-04-01T00:00:00", "price": 1482.37}, {"date": "2007-05-01T00:00:00", "price": 1530.62}, {"date": "2007-06-01T00:00:00", "price": 1503.35}, {"date": "2007-07-01T00:00:00", "price": 1455.27}, {"date": "2007-08-01T00:00:00", "price": 1473.99}, {"date": "2007-09-01T00:00:00", "price": 1526.75}, {"date": "2007-10-01T00:00:00", "price": 1549.38}, {"date": "2007-11-01T00:00:00", "price": 1481.14}, {"date": "2007-12-01T00:00:00", "price": 1468.36}, {"date": "2008-01-01T00:00:00", "price": 1378.55}, {"date": "2008-02-01T00:00:00", "price": 1330.63}, {"date": "2008-03-01T00:00:00", "price": 1322.7}, {"date": "2008-04-01T00:00:00", "price": 1385.59}, {"date": "2008-05-01T00:00:00", "price": 1400.38}, {"date": "2008-06-01T00:00:00", "price": 1280.0}, {"date": "2008-07-01T00:00:00", "price": 1267.38}, {"date": "2008-08-01T00:00:00", "price": 1282.83}, {"date": "2008-09-01T00:00:00", "price": 1166.36}, {"date": "2008-10-01T00:00:00", "price": 968.75}, {"date": "2008-11-01T00:00:00", "price": 896.24}, {"date": "2008-12-01T00:00:00", "price": 903.25}, {"date": "2009-01-01T00:00:00", "price": 825.88}, {"date": "2009-02-01T00:00:00", "price": 735.09}, {"date": "2009-03-01T00:00:00", "price": 797.87}, {"date": "2009-04-01T00:00:00", "price": 872.81}, {"date": "2009-05-01T00:00:00", "price": 919.14}, {"date": "2009-06-01T00:00:00", "price": 919.32}, {"date": "2009-07-01T00:00:00", "price": 987.48}, {"date": "2009-08-01T00:00:00", "price": 1020.62}, {"date": "2009-09-01T00:00:00", "price": 1057.08}, {"date": "2009-10-01T00:00:00", "price": 1036.19}, {"date": "2009-11-01T00:00:00", "price": 1095.63}, {"date": "2009-12-01T00:00:00", "price": 1115.1}, {"date": "2010-01-01T00:00:00", "price": 1073.87}, {"date": "2010-02-01T00:00:00", "price": 1104.49}, {"date": "2010-03-01T00:00:00", "price": 1140.45}]}}, {"mode": "vega-lite"}); </script>

{% endraw %}

The original distribution of prices.

{% raw %}

alt.Chart(sp500).mark_bar().encode(
    x=alt.X('price:Q', bin=alt.BinParams(maxbins=20)),
    y=alt.Y('count()'), tooltip=alt.Tooltip(['count()'])
)

</div>

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})({"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}}, "data": {"name": "data-776d9ed0e2795915ad0b63d5871c999b"}, "mark": "bar", "encoding": {"tooltip": [{"type": "quantitative", "aggregate": "count"}], "x": {"type": "quantitative", "bin": {"maxbins": 20}, "field": "price"}, "y": {"type": "quantitative", "aggregate": "count"}}, "$schema": "https://vega.github.io/schema/vega-lite/v4.8.1.json", "datasets": {"data-776d9ed0e2795915ad0b63d5871c999b": [{"date": "2000-01-01T00:00:00", "price": 1394.46}, {"date": "2000-02-01T00:00:00", "price": 1366.42}, {"date": "2000-03-01T00:00:00", "price": 1498.58}, {"date": "2000-04-01T00:00:00", "price": 1452.43}, {"date": "2000-05-01T00:00:00", "price": 1420.6}, {"date": "2000-06-01T00:00:00", "price": 1454.6}, {"date": "2000-07-01T00:00:00", "price": 1430.83}, {"date": "2000-08-01T00:00:00", "price": 1517.68}, {"date": "2000-09-01T00:00:00", "price": 1436.51}, {"date": "2000-10-01T00:00:00", "price": 1429.4}, {"date": "2000-11-01T00:00:00", "price": 1314.95}, {"date": "2000-12-01T00:00:00", "price": 1320.28}, {"date": "2001-01-01T00:00:00", "price": 1366.01}, {"date": "2001-02-01T00:00:00", "price": 1239.94}, {"date": "2001-03-01T00:00:00", "price": 1160.33}, {"date": "2001-04-01T00:00:00", "price": 1249.46}, {"date": "2001-05-01T00:00:00", "price": 1255.82}, {"date": "2001-06-01T00:00:00", "price": 1224.38}, {"date": "2001-07-01T00:00:00", "price": 1211.23}, {"date": "2001-08-01T00:00:00", "price": 1133.58}, {"date": "2001-09-01T00:00:00", "price": 1040.94}, {"date": "2001-10-01T00:00:00", "price": 1059.78}, {"date": "2001-11-01T00:00:00", "price": 1139.45}, {"date": "2001-12-01T00:00:00", "price": 1148.08}, {"date": "2002-01-01T00:00:00", "price": 1130.2}, {"date": "2002-02-01T00:00:00", "price": 1106.73}, {"date": "2002-03-01T00:00:00", "price": 1147.39}, {"date": "2002-04-01T00:00:00", "price": 1076.92}, {"date": "2002-05-01T00:00:00", "price": 1067.14}, {"date": "2002-06-01T00:00:00", "price": 989.82}, {"date": "2002-07-01T00:00:00", "price": 911.62}, {"date": "2002-08-01T00:00:00", "price": 916.07}, {"date": "2002-09-01T00:00:00", "price": 815.28}, {"date": "2002-10-01T00:00:00", "price": 885.76}, {"date": "2002-11-01T00:00:00", "price": 936.31}, {"date": "2002-12-01T00:00:00", "price": 879.82}, {"date": "2003-01-01T00:00:00", "price": 855.7}, {"date": "2003-02-01T00:00:00", "price": 841.15}, {"date": "2003-03-01T00:00:00", "price": 848.18}, {"date": "2003-04-01T00:00:00", "price": 916.92}, {"date": "2003-05-01T00:00:00", "price": 963.59}, {"date": "2003-06-01T00:00:00", "price": 974.5}, {"date": "2003-07-01T00:00:00", "price": 990.31}, {"date": "2003-08-01T00:00:00", "price": 1008.01}, {"date": "2003-09-01T00:00:00", "price": 995.97}, {"date": "2003-10-01T00:00:00", "price": 1050.71}, {"date": "2003-11-01T00:00:00", "price": 1058.2}, {"date": "2003-12-01T00:00:00", "price": 1111.92}, {"date": "2004-01-01T00:00:00", "price": 1131.13}, {"date": "2004-02-01T00:00:00", "price": 1144.94}, {"date": "2004-03-01T00:00:00", "price": 1126.21}, {"date": "2004-04-01T00:00:00", "price": 1107.3}, {"date": "2004-05-01T00:00:00", "price": 1120.68}, {"date": "2004-06-01T00:00:00", "price": 1140.84}, {"date": "2004-07-01T00:00:00", "price": 1101.72}, {"date": "2004-08-01T00:00:00", "price": 1104.24}, {"date": "2004-09-01T00:00:00", "price": 1114.58}, {"date": "2004-10-01T00:00:00", "price": 1130.2}, {"date": "2004-11-01T00:00:00", "price": 1173.82}, {"date": "2004-12-01T00:00:00", "price": 1211.92}, {"date": "2005-01-01T00:00:00", "price": 1181.27}, {"date": "2005-02-01T00:00:00", "price": 1203.6}, {"date": "2005-03-01T00:00:00", "price": 1180.59}, {"date": "2005-04-01T00:00:00", "price": 1156.85}, {"date": "2005-05-01T00:00:00", "price": 1191.5}, {"date": "2005-06-01T00:00:00", "price": 1191.33}, {"date": "2005-07-01T00:00:00", "price": 1234.18}, {"date": "2005-08-01T00:00:00", "price": 1220.33}, {"date": "2005-09-01T00:00:00", "price": 1228.81}, {"date": "2005-10-01T00:00:00", "price": 1207.01}, {"date": "2005-11-01T00:00:00", "price": 1249.48}, {"date": "2005-12-01T00:00:00", "price": 1248.29}, {"date": "2006-01-01T00:00:00", "price": 1280.08}, {"date": "2006-02-01T00:00:00", "price": 1280.66}, {"date": "2006-03-01T00:00:00", "price": 1294.87}, {"date": "2006-04-01T00:00:00", "price": 1310.61}, {"date": "2006-05-01T00:00:00", "price": 1270.09}, {"date": "2006-06-01T00:00:00", "price": 1270.2}, {"date": "2006-07-01T00:00:00", "price": 1276.66}, {"date": "2006-08-01T00:00:00", "price": 1303.82}, {"date": "2006-09-01T00:00:00", "price": 1335.85}, {"date": "2006-10-01T00:00:00", "price": 1377.94}, {"date": "2006-11-01T00:00:00", "price": 1400.63}, {"date": "2006-12-01T00:00:00", "price": 1418.3}, {"date": "2007-01-01T00:00:00", "price": 1438.24}, {"date": "2007-02-01T00:00:00", "price": 1406.82}, {"date": "2007-03-01T00:00:00", "price": 1420.86}, {"date": "2007-04-01T00:00:00", "price": 1482.37}, {"date": "2007-05-01T00:00:00", "price": 1530.62}, {"date": "2007-06-01T00:00:00", "price": 1503.35}, {"date": "2007-07-01T00:00:00", "price": 1455.27}, {"date": "2007-08-01T00:00:00", "price": 1473.99}, {"date": "2007-09-01T00:00:00", "price": 1526.75}, {"date": "2007-10-01T00:00:00", "price": 1549.38}, {"date": "2007-11-01T00:00:00", "price": 1481.14}, {"date": "2007-12-01T00:00:00", "price": 1468.36}, {"date": "2008-01-01T00:00:00", "price": 1378.55}, {"date": "2008-02-01T00:00:00", "price": 1330.63}, {"date": "2008-03-01T00:00:00", "price": 1322.7}, {"date": "2008-04-01T00:00:00", "price": 1385.59}, {"date": "2008-05-01T00:00:00", "price": 1400.38}, {"date": "2008-06-01T00:00:00", "price": 1280.0}, {"date": "2008-07-01T00:00:00", "price": 1267.38}, {"date": "2008-08-01T00:00:00", "price": 1282.83}, {"date": "2008-09-01T00:00:00", "price": 1166.36}, {"date": "2008-10-01T00:00:00", "price": 968.75}, {"date": "2008-11-01T00:00:00", "price": 896.24}, {"date": "2008-12-01T00:00:00", "price": 903.25}, {"date": "2009-01-01T00:00:00", "price": 825.88}, {"date": "2009-02-01T00:00:00", "price": 735.09}, {"date": "2009-03-01T00:00:00", "price": 797.87}, {"date": "2009-04-01T00:00:00", "price": 872.81}, {"date": "2009-05-01T00:00:00", "price": 919.14}, {"date": "2009-06-01T00:00:00", "price": 919.32}, {"date": "2009-07-01T00:00:00", "price": 987.48}, {"date": "2009-08-01T00:00:00", "price": 1020.62}, {"date": "2009-09-01T00:00:00", "price": 1057.08}, {"date": "2009-10-01T00:00:00", "price": 1036.19}, {"date": "2009-11-01T00:00:00", "price": 1095.63}, {"date": "2009-12-01T00:00:00", "price": 1115.1}, {"date": "2010-01-01T00:00:00", "price": 1073.87}, {"date": "2010-02-01T00:00:00", "price": 1104.49}, {"date": "2010-03-01T00:00:00", "price": 1140.45}]}}, {"mode": "vega-lite"}); </script>

{% endraw %}

We get the standardized value by dividing the values' difference from the mean by the standard deviation.

$ \text{standardized value} = \frac { \text{value} - \text{mean}} {\text{standard deviation}} $

The mean and standard deviation of the prices

{% raw %}

print(f"""
mean: {sp500['price'].mean():.2f}
std: {sp500['price'].std():.2f}
""")

</div>

mean: 1184.43
std: 195.41

{% endraw %}

We standardize the data set.

{% raw %}

standardized_prices = sp500.copy()
standardized_prices['price'] = (sp500['price'] - sp500['price'].mean()) / sp500['price'].std()
standardized_prices

</div>

<style scoped> .dataframe tbody tr th:only-of-type { vertical-align: middle; }

.dataframe tbody tr th {
    vertical-align: top;
}

.dataframe thead th {
    text-align: right;
}

</style>

	date	price
0	2000-01-01	1.074812
1	2000-02-01	0.931317
2	2000-03-01	1.607646
3	2000-04-01	1.371473
4	2000-05-01	1.208583
...	...	...
118	2009-11-01	-0.454452
119	2009-12-01	-0.354814
120	2010-01-01	-0.565809
121	2010-02-01	-0.409111
122	2010-03-01	-0.225086

123 rows × 2 columns

{% endraw %}

{% raw %}

alt.Chart(standardized_prices).mark_bar().encode(
    x=alt.X('price:Q', bin=alt.BinParams(maxbins=20)),
    y=alt.Y('count()'), tooltip=alt.Tooltip(['count()'])
)

</div>

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function loadScript(lib) {
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    s.src = paths[lib];
    s.async = true;
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    s.onerror = () => reject(`Error loading script: ${paths[lib]}`);
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}

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  outputDiv.innerHTML = `<div class="error" style="color:red;">${err}</div>`;
  throw err;
}

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    .catch(err => showError(`Javascript Error: ${err.message}<br>This usually means there's a typo in your chart specification. See the javascript console for the full traceback.`));
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    .then(() => loadScript("vega-lite"))
    .then(() => loadScript("vega-embed"))
    .catch(showError)
    .then(() => displayChart(vegaEmbed));
}

})({"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}}, "data": {"name": "data-fd0ed95555efecbeec45d95260c94f8c"}, "mark": "bar", "encoding": {"tooltip": [{"type": "quantitative", "aggregate": "count"}], "x": {"type": "quantitative", "bin": {"maxbins": 20}, "field": "price"}, "y": {"type": "quantitative", "aggregate": "count"}}, "$schema": "https://vega.github.io/schema/vega-lite/v4.8.1.json", "datasets": {"data-fd0ed95555efecbeec45d95260c94f8c": [{"date": "2000-01-01T00:00:00", "price": 1.0748115732150199}, {"date": "2000-02-01T00:00:00", "price": 0.931316747897087}, {"date": "2000-03-01T00:00:00", "price": 1.6076461385539347}, {"date": "2000-04-01T00:00:00", "price": 1.3714733072977796}, {"date": "2000-05-01T00:00:00", "price": 1.2085831400413662}, {"date": "2000-06-01T00:00:00", "price": 1.3825782919960639}, {"date": "2000-07-01T00:00:00", "price": 1.2609352107618532}, {"date": "2000-08-01T00:00:00", "price": 1.7053904739167214}, {"date": "2000-09-01T00:00:00", "price": 1.2900026361472265}, {"date": "2000-10-01T00:00:00", "price": 1.2536171793708182}, {"date": "2000-11-01T00:00:00", "price": 0.6679187928644901}, {"date": "2000-12-01T00:00:00", "price": 0.6951950916856232}, {"date": "2001-01-01T00:00:00", "price": 0.9292185710646917}, {"date": "2001-02-01T00:00:00", "price": 0.2840547826256115}, {"date": "2001-03-01T00:00:00", "price": -0.1233497481718445}, {"date": "2001-04-01T00:00:00", "price": 0.3327734251729268}, {"date": "2001-05-01T00:00:00", "price": 0.36532075359739324}, {"date": "2001-06-01T00:00:00", "price": 0.20442641308399132}, {"date": "2001-07-01T00:00:00", "price": 0.1371312293132769}, {"date": "2001-08-01T00:00:00", "price": -0.2602429927244375}, {"date": "2001-09-01T00:00:00", "price": -0.7343286067562955}, {"date": "2001-10-01T00:00:00", "price": -0.6379148225555163}, {"date": "2001-11-01T00:00:00", "price": -0.23020324148990529}, {"date": "2001-12-01T00:00:00", "price": -0.18603917792022823}, {"date": "2002-01-01T00:00:00", "price": -0.2775401578305216}, {"date": "2002-02-01T00:00:00", "price": -0.39764798772395565}, {"date": "2002-03-01T00:00:00", "price": -0.18957025600401387}, {"date": "2002-04-01T00:00:00", "price": -0.5502007959524712}, {"date": "2002-05-01T00:00:00", "price": -0.600249989661793}, {"date": "2002-06-01T00:00:00", "price": -0.9959354352246528}, {"date": "2002-07-01T00:00:00", "price": -1.3961242847204576}, {"date": "2002-08-01T00:00:00", "price": -1.3733513898322691}, {"date": "2002-09-01T00:00:00", "price": -1.889144665288563}, {"date": "2002-10-01T00:00:00", "price": -1.528462950295413}, {"date": "2002-11-01T00:00:00", "price": -1.269773099374532}, {"date": "2002-12-01T00:00:00", "price": -1.5588609268427924}, {"date": "2003-01-01T00:00:00", "price": -1.6822951346412427}, {"date": "2003-02-01T00:00:00", "price": -1.756754824668915}, {"date": "2003-03-01T00:00:00", "price": -1.7207787682500468}, {"date": "2003-04-01T00:00:00", "price": -1.3690015110334022}, {"date": "2003-05-01T00:00:00", "price": -1.130167577453233}, {"date": "2003-06-01T00:00:00", "price": -1.074335603693652}, {"date": "2003-07-01T00:00:00", "price": -0.9934278580347179}, {"date": "2003-08-01T00:00:00", "price": -0.9028480289288898}, {"date": "2003-09-01T00:00:00", "price": -0.9644627827387297}, {"date": "2003-10-01T00:00:00", "price": -0.6843305880916662}, {"date": "2003-11-01T00:00:00", "price": -0.6460004796169402}, {"date": "2003-12-01T00:00:00", "price": -0.3710881395285177}, {"date": "2004-01-01T00:00:00", "price": -0.2727808786741133}, {"date": "2004-02-01T00:00:00", "price": -0.20210814195369084}, {"date": "2004-03-01T00:00:00", "price": -0.2979590006628523}, {"date": "2004-04-01T00:00:00", "price": -0.3947310101764802}, {"date": "2004-05-01T00:00:00", "price": -0.32625880037783683}, {"date": "2004-06-01T00:00:00", "price": -0.22308991027764036}, {"date": "2004-07-01T00:00:00", "price": -0.4232866851149273}, {"date": "2004-08-01T00:00:00", "price": 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1.5246913911072977}, {"date": "2007-05-01T00:00:00", "price": 1.7716109817488908}, {"date": "2007-06-01T00:00:00", "price": 1.6320566348722847}, {"date": "2007-07-01T00:00:00", "price": 1.3860070199904657}, {"date": "2007-08-01T00:00:00", "price": 1.4818067036549345}, {"date": "2007-09-01T00:00:00", "price": 1.7518062394528713}, {"date": "2007-10-01T00:00:00", "price": 1.867615365592131}, {"date": "2007-11-01T00:00:00", "price": 1.5183968606101141}, {"date": "2007-12-01T00:00:00", "price": 1.4529951534930239}, {"date": "2008-01-01T00:00:00", "price": 0.9933920771091594}, {"date": "2008-02-01T00:00:00", "price": 0.7481612629424216}, {"date": "2008-03-01T00:00:00", "price": 0.7075794525012227}, {"date": "2008-04-01T00:00:00", "price": 1.0294193085727201}, {"date": "2008-05-01T00:00:00", "price": 1.1051071996730146}, {"date": "2008-06-01T00:00:00", "price": 0.48906201166399915}, {"date": "2008-07-01T00:00:00", "price": 0.42447910526199134}, {"date": "2008-08-01T00:00:00", "price": 0.5035445493119928}, {"date": "2008-09-01T00:00:00", "price": -0.09249119622223208}, {"date": "2008-10-01T00:00:00", "price": -1.1037612543918731}, {"date": "2008-11-01T00:00:00", "price": -1.474831503457612}, {"date": "2008-12-01T00:00:00", "price": -1.4389577971281289}, {"date": "2009-01-01T00:00:00", "price": -1.8348991179144511}, {"date": "2009-02-01T00:00:00", "price": -2.2995173486781866}, {"date": "2009-03-01T00:00:00", "price": -1.9782404180983066}, {"date": "2009-04-01T00:00:00", "price": -1.594734633172276}, {"date": "2009-05-01T00:00:00", "price": -1.357640651111654}, {"date": "2009-06-01T00:00:00", "price": -1.3567195003071877}, {"date": "2009-07-01T00:00:00", "price": -1.0079103956827116}, {"date": "2009-08-01T00:00:00", "price": -0.8383162975715739}, {"date": "2009-09-01T00:00:00", "price": -0.6517320846225073}, {"date": "2009-10-01T00:00:00", "price": -0.7586367529852607}, {"date": "2009-11-01T00:00:00", "price": -0.45445228733269477}, {"date": "2009-12-01T00:00:00", "price": -0.35481447531628507}, {"date": "2010-01-01T00:00:00", "price": -0.5658091845837024}, {"date": "2010-02-01T00:00:00", "price": -0.40911119773508875}, {"date": "2010-03-01T00:00:00", "price": -0.2250857370206495}]}}, {"mode": "vega-lite"}); </script>

{% endraw %}

The new mean and standard deviation

{% raw %}

print(f"""
mean: {standardized_prices['price'].mean():.2f}
std: {standardized_prices['price'].std():.2f}
""")

</div>

mean: -0.00
std: 1.00

{% endraw %}

When plotted on the dates, it provides a similar graph as before, but now with a value domain around 0.

{% raw %}

alt.Chart(standardized_prices).mark_line(point=True, strokeWidth=1).encode(
    alt.X('date:T'), alt.Y('price'), alt.Tooltip('price')
).configure_point(size=10)

</div>

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} else {
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    .then(() => loadScript("vega-lite"))
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})({"config": {"view": {"continuousWidth": 400, "continuousHeight": 300}, "point": {"size": 10}}, "data": {"name": "data-fd0ed95555efecbeec45d95260c94f8c"}, "mark": {"type": "line", "point": true, "strokeWidth": 1}, "encoding": {"tooltip": {"type": "quantitative", "field": "price"}, "x": {"type": "temporal", "field": "date"}, "y": {"type": "quantitative", "field": "price"}}, "$schema": "https://vega.github.io/schema/vega-lite/v4.8.1.json", "datasets": {"data-fd0ed95555efecbeec45d95260c94f8c": [{"date": "2000-01-01T00:00:00", "price": 1.0748115732150199}, {"date": "2000-02-01T00:00:00", "price": 0.931316747897087}, {"date": "2000-03-01T00:00:00", "price": 1.6076461385539347}, {"date": "2000-04-01T00:00:00", "price": 1.3714733072977796}, {"date": "2000-05-01T00:00:00", "price": 1.2085831400413662}, {"date": "2000-06-01T00:00:00", "price": 1.3825782919960639}, {"date": "2000-07-01T00:00:00", "price": 1.2609352107618532}, {"date": "2000-08-01T00:00:00", "price": 1.7053904739167214}, 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