Z Table

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• Author: Diego Inácio
• GitHub: github.com/diegoinacio
• Notebook: z-table.ipynb

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Study about standard normal distribution.

%matplotlib inline
import matplotlib
import matplotlib.pyplot as plt
from matplotlib.patches import Polygon
from IPython.display import display, HTML

import numpy as np
import pandas as pd

np.seterr(divide='ignore')
plt.rcParams['figure.figsize'] = (16, 8)

Normal distribution

---

$$ \large f(x|\mu, \sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}e^{-\frac{(x-\mu)^2}{2\sigma^2}}=\frac{1}{\sigma\sqrt{2\pi}}e^{-\frac{1}{2}\left(\frac{x-\mu}{\sigma}\right)^2} $$

where,

• $\mu$ is the mean of the distribution;
• $\sigma^2$ is the variance;
• $\sigma$ is the standard deviation.

def normalDistribution(x, mu, sigma):
    return 1/(sigma*(2*np.pi)**0.5)*np.exp(-1/2*((x - mu)/sigma)**2)

x = np.linspace(-4, 4, 101)
p = normalDistribution(x, 0, 1)

plt.plot(x, p)
plt.ylim(0, 1)

plt.annotate(
    "",
    xy=(-1, 0.5), xycoords='data',
    xytext=(1, 0.5), textcoords='data',
    arrowprops=dict(arrowstyle="<|-|>")
)
plt.text(0, 0.51, '68% within\n1 standard deviation', ha='center', va='bottom', size=12)

plt.annotate(
    "",
    xy=(-2, 0.65), xycoords='data',
    xytext=(2, 0.65), textcoords='data',
    arrowprops=dict(arrowstyle="<|-|>")
)
plt.text(0, 0.66, '95% within\n2 standard deviation', ha='center', va='bottom', size=12)

plt.annotate(
    "",
    xy=(-3, 0.8), xycoords='data',
    xytext=(3, 0.8), textcoords='data',
    arrowprops=dict(arrowstyle="<|-|>")
)
plt.text(0, 0.81, '99.7% within\n3 standard deviation', ha='center', va='bottom', size=12)

plt.grid(alpha=0.25, linestyle='-.')
plt.xticks(
    range(-4, 5), 
    [
        '', r'$\mu-3\sigma$', r'$\mu-2\sigma$', 
        r'$\mu-\sigma$', r'$\mu$', r'$\mu+\sigma$', 
        r'$\mu+2\sigma$', r'$\mu+3\sigma$', ''
    ]
)
plt.yticks(np.linspace(0, 1, 6), ['']*6)
plt.title('Probability density')
plt.show()

Standard normal distribution

---

Special case of density probability, when $\mu=0$ and $\sigma=1$ $$ \large f(x)=\frac{1}{\sqrt{2\pi}}e^{-\frac{1}{2}x^2} $$

def stdNormalDistribution(x):
    return normalDistribution(x, 0, 1)

Cumulative distribution function

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$$ \large \Phi(z)=\frac{1}{2\pi}\int_{-\infty}^{z}e^{-\frac{t^2}{2}}dx $$

For the range $[-\infty, \infty]$ the value of $\Phi$ should be $1$.

fig, ax = plt.subplots()

plt.plot(x, p, c='red', linewidth=4)
x1 = x[:65]; y1 = p[:65]
verts = [(x1[0], 0), *zip(x1, y1), (x1[-1], 0)]
P = Polygon(verts, facecolor='0.8', edgecolor='0.6', hatch='x')
ax.add_patch(P)
plt.xlim([-4, 4])
plt.grid()
plt.show()

Cumulative from mean

---

$$ \large f(z)=\frac{1}{2\pi}\int_{0}^{z}e^{-\frac{t^2}{2}}dx $$

Probability from 0 to Z, or: $$ \large f(z)=\Phi(z) - \frac{1}{2} $$

fig, ax = plt.subplots()

plt.plot(x, p, c='red', linewidth=4)
x1 = x[50:65]; y1 = p[50:65]
verts = [(x1[0], 0), *zip(x1, y1), (x1[-1], 0)]
P = Polygon(verts, facecolor='0.8', edgecolor='0.6', hatch='x')
ax.add_patch(P)
plt.xlim([-4, 4])
plt.grid()
plt.show()

Z-Score

---

def definiteIntegral(f, a, b, N=10000):
    result = 0
    dx = abs(b - a)/N
    while a < b:
        result += f(a + dx/2)*dx
        a += dx
    return result

def zScore(z, option=0):
    '''
    Cumulative standard normal distribution
    Option:
        0 => one-tailed under *default
        1 => one-tailed above
        2 => two-tailed inside
        3 => two-tailed outside
    '''
    Z = abs(z)
    try:
        sign = z/abs(z)
    except ZeroDivisionError:
        return 0.0
    
    p = definiteIntegral(stdNormalDistribution, 0, Z)

    if option == 1:
        return 0.5 - p*sign

    if option == 2:
        return 2*p

    if option == 3:
        return 1 - 2*p

    return 0.5 + p*sign

fig, [[axA, axB], [axC, axD]] = plt.subplots(2, 2, sharex='col', sharey='row')

axA.plot(x, p, c='red', linewidth=2)
slice = x <= 1.5
xP = x[slice]; yP = p[slice]
vertsP = [(xP[0], 0), *zip(xP, yP), (xP[-1], 0)]
PP = Polygon(vertsP, facecolor='0.8', edgecolor='0.7', hatch='xx')
axA.add_patch(PP)
axA.set_title('one-tailed under')

axB.plot(x, p, c='red', linewidth=2)
slice = x >= 1.5
xP = x[slice]; yP = p[slice]
vertsP = [(xP[0], 0), *zip(xP, yP), (xP[-1], 0)]
PP = Polygon(vertsP, facecolor='0.8', edgecolor='0.7', hatch='xx')
axB.add_patch(PP)
axB.set_title('one-tailed above')

axC.plot(x, p, c='red', linewidth=2)
slice = abs(x) <= 1.5
xP = x[slice]; yP = p[slice]
vertsP = [(xP[0], 0), *zip(xP, yP), (xP[-1], 0)]
PP = Polygon(vertsP, facecolor='0.8', edgecolor='0.7', hatch='xx')
axC.add_patch(PP)
axC.set_title('two-tailed inside')

axD.plot(x, p, c='red', linewidth=2)
slice = x >= 1.5
xP = x[slice]; yP = p[slice]
vertsP = [(xP[0], 0), *zip(xP, yP), (xP[-1], 0)]
PP = Polygon(vertsP, facecolor='0.8', edgecolor='0.7', hatch='xx')
axD.add_patch(PP)
slice = x <= -1.5
xP = x[slice]; yP = p[slice]
vertsP = [(xP[0], 0), *zip(xP, yP), (xP[-1], 0)]
PP = Polygon(vertsP, facecolor='0.8', edgecolor='0.7', hatch='xx')
axD.add_patch(PP)
axD.set_title('two-tailed outside')

plt.show()

def zTable(option=0, digits=5, hsteps=0.01, zmin=-3.4, zmax=3.4):
    if option >= 2:
        zmin = 0.0 if zmin <= 0 else zmin
    steps = np.arange(zmin, zmax + 0.1, hsteps)
    steps = steps.reshape(-1, int(0.1/hsteps))
    df = pd.DataFrame(data=steps)
    d = int(np.ceil(abs(np.log10(hsteps))))
    cols = ['{0:.{1}f}'.format(e, d) for e in np.arange(0, 0.1, hsteps)]
    idxs = np.linspace(zmin, zmax, int((zmax - zmin)/0.1) + 1)
    df.index = idxs; df.columns = cols; ztable=df.rename_axis('+=>', axis=1)
    ztable = ztable.applymap(lambda x: round(zScore(x, option=option), digits))
    return ztable

# Display z table
ztable = zTable(zmin=1, zmax=3)
with pd.option_context('display.max_rows', None, 'display.max_columns', None):
    display(ztable)

+=>	0.00	0.01	0.02	0.03	0.04	0.05	0.06	0.07	0.08	0.09
1.0	0.84137	0.84375	0.84614	0.84852	0.85085	0.85314	0.85543	0.85771	0.85995	0.86214
1.1	0.86433	0.86652	0.86867	0.87076	0.87288	0.87495	0.87698	0.87902	0.88102	0.88300
1.2	0.88495	0.88686	0.88879	0.89065	0.89251	0.89437	0.89617	0.89798	0.89975	0.90147
1.3	0.90322	0.90490	0.90658	0.90824	0.90988	0.91151	0.91311	0.91466	0.91623	0.91776
1.4	0.91924	0.92075	0.92220	0.92364	0.92509	0.92647	0.92788	0.92924	0.93056	0.93191
1.5	0.93319	0.93448	0.93576	0.93699	0.93824	0.93945	0.94062	0.94181	0.94295	0.94408
1.6	0.94522	0.94630	0.94740	0.94847	0.94950	0.95055	0.95154	0.95254	0.95354	0.95449
1.7	0.95543	0.95638	0.95728	0.95820	0.95909	0.95994	0.96081	0.96164	0.96246	0.96329
1.8	0.96407	0.96487	0.96563	0.96638	0.96713	0.96784	0.96856	0.96927	0.96995	0.97063
1.9	0.97130	0.97193	0.97258	0.97320	0.97381	0.97442	0.97500	0.97558	0.97616	0.97670
2.0	0.97726	0.97780	0.97831	0.97883	0.97932	0.97982	0.98031	0.98077	0.98125	0.98170
2.1	0.98214	0.98258	0.98300	0.98341	0.98383	0.98422	0.98462	0.98500	0.98537	0.98575
2.2	0.98610	0.98645	0.98680	0.98713	0.98746	0.98778	0.98809	0.98840	0.98870	0.98899
2.3	0.98928	0.98956	0.98983	0.99010	0.99036	0.99062	0.99087	0.99111	0.99135	0.99158
2.4	0.99181	0.99203	0.99224	0.99245	0.99266	0.99286	0.99305	0.99324	0.99343	0.99361
2.5	0.99379	0.99396	0.99413	0.99430	0.99446	0.99462	0.99477	0.99492	0.99506	0.99520
2.6	0.99534	0.99547	0.99560	0.99573	0.99585	0.99598	0.99609	0.99621	0.99632	0.99643
2.7	0.99654	0.99664	0.99674	0.99684	0.99693	0.99702	0.99711	0.99720	0.99728	0.99737
2.8	0.99744	0.99752	0.99760	0.99767	0.99774	0.99782	0.99788	0.99795	0.99801	0.99808
2.9	0.99813	0.99819	0.99825	0.99831	0.99836	0.99841	0.99846	0.99851	0.99856	0.99861
3.0	0.99865	0.99870	0.99874	0.99878	0.99882	0.99886	0.99889	0.99893	0.99897	0.99900

Export HTML table

---

Access the extended Z-Table

%%time
# save z table as HTML
HTML = '''
<!DOCTYPE html>
<html>
<head>
  <link rel="stylesheet" href="z-table.css">
</head>
<body>
'''
HTML += '\n<h1>Z Tables</h1>\n'

HTML += '\n<h2>One-tailed under</h2>\n'
ztable = zTable(option=0, hsteps=0.005)
HTML += ztable.to_html()

HTML += '\n<h2>One-tailed above</h2>\n'
ztable = zTable(option=1, hsteps=0.005)
HTML += ztable.to_html()

HTML += '\n<h2>Two-tailed inside</h2>\n'
ztable = zTable(option=2, hsteps=0.005)
HTML += ztable.to_html()

HTML += '\n<h2>Two-tailed outside</h2>\n'
ztable = zTable(option=3, hsteps=0.005)
HTML += ztable.to_html()

HTML += '''
</body>
</html>
'''

ztable_html = open('output/z-table.html', 'w')
ztable_html.write(HTML)
ztable_html.close()

Wall time: 1min 14s