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Standard normal table
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<h2 id="normal-and-standard-normal-distribution">Normal and standard normal distribution</h2> <p><a href="/facts/Normal_distribution/UapjjPyQ">Normal distributions</a> are symmetrical, bell-shaped distributions that are useful in describing real-world data. The <i>standard</i> normal distribution, represented by Z, is the normal distribution having a <a href="/facts/Mean/swcEd4Pg">mean</a> of 0 and a <a href="/facts/Standard_deviation/qSug9BRl">standard deviation</a> of 1. </p> <h3>Conversion</h3> <p class="note">Main article: <a href="/facts/Standard_normal_deviate/eLjQKyPn">Standard normal deviate</a></p> <p>If X is a random variable from a normal distribution with mean μ and standard deviation σ, its <a href="/facts/Z-score/lF8v0Mcy">Z-score</a> may be calculated from X by subtracting μ and dividing by the standard deviation: </p> Z = X − μ σ {\displaystyle Z={\frac {X-\mu }{\sigma }}} <p>If X ¯ {\displaystyle {\overline {X}}} is the mean of a sample of size n from some population in which the mean is μ and the standard deviation is σ, the standard error is σ n : {\displaystyle {\tfrac {\sigma }{\sqrt {n}}}:} </p> Z = X ¯ − μ σ / n {\displaystyle Z={\frac {{\overline {X}}-\mu }{\sigma /{\sqrt {n}}}}} <p>If ∑ X {\textstyle \sum X} is the total of a sample of size n from some population in which the mean is μ and the standard deviation is σ, the expected total is nμ and the standard error is σ n : {\displaystyle \sigma {\sqrt {n}}:} </p> Z = ∑ X − n μ σ n {\displaystyle Z={\frac {\sum {X}-n\mu }{\sigma {\sqrt {n}}}}} <h2 id="reading-a-z-table">Reading a Z table</h2> <h3>Formatting / layout</h3> <p>Z tables are typically composed as follows: </p> <ul><li>The label for rows contains the integer part and the first decimal place of Z.</li> <li>The label for columns contains the second decimal place of Z.</li> <li>The values within the table are the probabilities corresponding to the table type. These probabilities are calculations of the area under the normal curve from the starting point (0 for <i>cumulative from mean</i>, negative infinity for <i>cumulative</i> and positive infinity for <i>complementary cumulative</i>) to Z.</li></ul> <p>Example: To find 0.69, one would look down the rows to find 0.6 and then across the columns to 0.09 which would yield a probability of 0.25490 for a <i>cumulative from mean</i> table or 0.75490 from a <i>cumulative</i> table. </p><p>To find a negative value such as –0.83, one could use a <i>cumulative</i> table for negative z-values<a class="footnote-ref" id="fnref:3" href="#fn:3"><sup>3</sup></a> which yield a probability of 0.20327. </p><p>But since the normal distribution curve is symmetrical, probabilities for only positive values of Z are typically given. The user might have to use a complementary operation on the absolute value of Z, as in the example below. </p> <h3>Types of tables</h3> <p>Z tables use at least three different conventions: </p> Cumulative from mean gives a probability that a statistic is between 0 (mean) and Z. Example: Prob(0 ≤ Z ≤ 0.69) = 0.2549. Cumulative gives a probability that a statistic is less than Z. This equates to the area of the distribution below Z. Example: Prob(Z ≤ 0.69) = 0.7549. Complementary cumulative gives a probability that a statistic is greater than Z. This equates to the area of the distribution above Z. Example: Find Prob(<i>Z</i> ≥ 0.69). Since this is the portion of the area above Z, the proportion that is greater than Z is found by subtracting Z from 1. That is Prob(<i>Z</i> ≥ 0.69) = 1 − Prob(Z ≤ 0.69) or Prob(<i>Z</i> ≥ 0.69) = 1 − 0.7549 = 0.2451. <h2 id="table-examples">Table examples</h2> <h3>Cumulative from minus infinity to Z</h3> <p>This table gives a probability that a statistic is between minus infinity and Z. </p> f ( z ) = Φ ( z ) {\displaystyle f(z)=\Phi (z)} <p>The values are calculated using the <a href="/facts/Normal_cumulative_distribution_function/UapjjPyQ">cumulative distribution function</a> of a standard normal distribution with mean of zero and standard deviation of one, usually denoted with the capital Greek letter Φ {\displaystyle \Phi } (<a href="/facts/Phi_(letter)/pUzerkKk">phi</a>), is the integral </p> Φ ( z ) = 1 2 π ∫ − ∞ z e − t 2 / 2 d t {\displaystyle \Phi (z)={\frac {1}{\sqrt {2\pi }}}\int _{-\infty }^{z}e^{-t^{2}/2}\,dt} <p> Φ {\displaystyle \Phi } (z) is related to the <a href="/facts/Error_function/Ca2H7pGr">error function</a>, or erf(<i>z</i>). </p> Φ ( z ) = 1 2 [ 1 + erf ( z 2 ) ] {\displaystyle \Phi (z)={\frac {1}{2}}\left[1+\operatorname {erf} \left({\frac {z}{\sqrt {2}}}\right)\right]} <p>Note that for <i>z</i> = 1, 2, 3, one obtains (after multiplying by 2 to account for the [−<i>z</i>,<i>z</i>] interval) the results <i>f</i> (<i>z</i>) = 0.6827, 0.9545, 0.9974, characteristic of the <a href="/facts/68%25E2%2580%259395%25E2%2580%259399.7_rule/IDaupJFh">68–95–99.7 rule</a>. </p> <h3>Cumulative (less than Z)</h3> <p>This table gives a probability that a statistic is less than Z (i.e. between negative infinity and Z). </p> <table><tbody><tr><th><i>z</i></th><th>−0.00</th><th>−0.01</th><th>−0.02</th><th>−0.03</th><th>−0.04</th><th>−0.05</th><th>−0.06</th><th>−0.07</th><th>−0.08</th><th>−0.09</th></tr><tr><th>-3.9</th><td>0.00005</td><td>0.00005</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00003</td><td>0.00003</td></tr><tr><th>-3.8</th><td>0.00007</td><td>0.00007</td><td>0.00007</td><td>0.00006</td><td>0.00006</td><td>0.00006</td><td>0.00006</td><td>0.00005</td><td>0.00005</td><td>0.00005</td></tr><tr><th>-3.7</th><td>0.00011</td><td>0.00010</td><td>0.00010</td><td>0.00010</td><td>0.00009</td><td>0.00009</td><td>0.00008</td><td>0.00008</td><td>0.00008</td><td>0.00008</td></tr><tr><th>-3.6</th><td>0.00016</td><td>0.00015</td><td>0.00015</td><td>0.00014</td><td>0.00014</td><td>0.00013</td><td>0.00013</td><td>0.00012</td><td>0.00012</td><td>0.00011</td></tr><tr><th>-3.5</th><td>0.00023</td><td>0.00022</td><td>0.00022</td><td>0.00021</td><td>0.00020</td><td>0.00019</td><td>0.00019</td><td>0.00018</td><td>0.00017</td><td>0.00017</td></tr><tr><th>-3.4</th><td>0.00034</td><td>0.00032</td><td>0.00031</td><td>0.00030</td><td>0.00029</td><td>0.00028</td><td>0.00027</td><td>0.00026</td><td>0.00025</td><td>0.00024</td></tr><tr><th>-3.3</th><td>0.00048</td><td>0.00047</td><td>0.00045</td><td>0.00043</td><td>0.00042</td><td>0.00040</td><td>0.00039</td><td>0.00038</td><td>0.00036</td><td>0.00035</td></tr><tr><th>-3.2</th><td>0.00069</td><td>0.00066</td><td>0.00064</td><td>0.00062</td><td>0.00060</td><td>0.00058</td><td>0.00056</td><td>0.00054</td><td>0.00052</td><td>0.00050</td></tr><tr><th>-3.1</th><td>0.00097</td><td>0.00094</td><td>0.00090</td><td>0.00087</td><td>0.00084</td><td>0.00082</td><td>0.00079</td><td>0.00076</td><td>0.00074</td><td>0.00071</td></tr><tr><th>-3.0</th><td>0.00135</td><td>0.00131</td><td>0.00126</td><td>0.00122</td><td>0.00118</td><td>0.00114</td><td>0.00111</td><td>0.00107</td><td>0.00104</td><td>0.00100</td></tr><tr><th>-2.9</th><td>0.00187</td><td>0.00181</td><td>0.00175</td><td>0.00169</td><td>0.00164</td><td>0.00159</td><td>0.00154</td><td>0.00149</td><td>0.00144</td><td>0.00139</td></tr><tr><th>-2.8</th><td>0.00256</td><td>0.00248</td><td>0.00240</td><td>0.00233</td><td>0.00226</td><td>0.00219</td><td>0.00212</td><td>0.00205</td><td>0.00199</td><td>0.00193</td></tr><tr><th>-2.7</th><td>0.00347</td><td>0.00336</td><td>0.00326</td><td>0.00317</td><td>0.00307</td><td>0.00298</td><td>0.00289</td><td>0.00280</td><td>0.00272</td><td>0.00264</td></tr><tr><th>-2.6</th><td>0.00466</td><td>0.00453</td><td>0.00440</td><td>0.00427</td><td>0.00415</td><td>0.00402</td><td>0.00391</td><td>0.00379</td><td>0.00368</td><td>0.00357</td></tr><tr><th>-2.5</th><td>0.00621</td><td>0.00604</td><td>0.00587</td><td>0.00570</td><td>0.00554</td><td>0.00539</td><td>0.00523</td><td>0.00508</td><td>0.00494</td><td>0.00480</td></tr><tr><th>-2.4</th><td>0.00820</td><td>0.00798</td><td>0.00776</td><td>0.00755</td><td>0.00734</td><td>0.00714</td><td>0.00695</td><td>0.00676</td><td>0.00657</td><td>0.00639</td></tr><tr><th>-2.3</th><td>0.01072</td><td>0.01044</td><td>0.01017</td><td>0.00990</td><td>0.00964</td><td>0.00939</td><td>0.00914</td><td>0.00889</td><td>0.00866</td><td>0.00842</td></tr><tr><th>-2.2</th><td>0.01390</td><td>0.01355</td><td>0.01321</td><td>0.01287</td><td>0.01255</td><td>0.01222</td><td>0.01191</td><td>0.01160</td><td>0.01130</td><td>0.01101</td></tr><tr><th>-2.1</th><td>0.01786</td><td>0.01743</td><td>0.01700</td><td>0.01659</td><td>0.01618</td><td>0.01578</td><td>0.01539</td><td>0.01500</td><td>0.01463</td><td>0.01426</td></tr><tr><th>-2.0</th><td>0.02275</td><td>0.02222</td><td>0.02169</td><td>0.02118</td><td>0.02068</td><td>0.02018</td><td>0.01970</td><td>0.01923</td><td>0.01876</td><td>0.01831</td></tr><tr><th>−1.9</th><td>0.02872</td><td>0.02807</td><td>0.02743</td><td>0.02680</td><td>0.02619</td><td>0.02559</td><td>0.02500</td><td>0.02442</td><td>0.02385</td><td>0.02330</td></tr><tr><th>−1.8</th><td>0.03593</td><td>0.03515</td><td>0.03438</td><td>0.03362</td><td>0.03288</td><td>0.03216</td><td>0.03144</td><td>0.03074</td><td>0.03005</td><td>0.02938</td></tr><tr><th>−1.7</th><td>0.04457</td><td>0.04363</td><td>0.04272</td><td>0.04182</td><td>0.04093</td><td>0.04006</td><td>0.03920</td><td>0.03836</td><td>0.03754</td><td>0.03673</td></tr><tr><th>−1.6</th><td>0.05480</td><td>0.05370</td><td>0.05262</td><td>0.05155</td><td>0.05050</td><td>0.04947</td><td>0.04846</td><td>0.04746</td><td>0.04648</td><td>0.04551</td></tr><tr><th>−1.5</th><td>0.06681</td><td>0.06552</td><td>0.06426</td><td>0.06301</td><td>0.06178</td><td>0.06057</td><td>0.05938</td><td>0.05821</td><td>0.05705</td><td>0.05592</td></tr><tr><td colspan="11"></td></tr><tr><th>−1.4</th><td>0.08076</td><td>0.07927</td><td>0.07780</td><td>0.07636</td><td>0.07493</td><td>0.07353</td><td>0.07215</td><td>0.07078</td><td>0.06944</td><td>0.06811</td></tr><tr><th>−1.3</th><td>0.09680</td><td>0.09510</td><td>0.09342</td><td>0.09176</td><td>0.09012</td><td>0.08851</td><td>0.08692</td><td>0.08534</td><td>0.08379</td><td>0.08226</td></tr><tr><th>−1.2</th><td>0.11507</td><td>0.11314</td><td>0.11123</td><td>0.10935</td><td>0.10749</td><td>0.10565</td><td>0.10383</td><td>0.10204</td><td>0.10027</td><td>0.09853</td></tr><tr><th>−1.1</th><td>0.13567</td><td>0.13350</td><td>0.13136</td><td>0.12924</td><td>0.12714</td><td>0.12507</td><td>0.12302</td><td>0.12100</td><td>0.11900</td><td>0.11702</td></tr><tr><th>−1.0</th><td>0.15866</td><td>0.15625</td><td>0.15386</td><td>0.15151</td><td>0.14917</td><td>0.14686</td><td>0.14457</td><td>0.14231</td><td>0.14007</td><td>0.13786</td></tr><tr><td colspan="11"></td></tr><tr><th>−0.9</th><td>0.18406</td><td>0.18141</td><td>0.17879</td><td>0.17619</td><td>0.17361</td><td>0.17106</td><td>0.16853</td><td>0.16602</td><td>0.16354</td><td>0.16109</td></tr><tr><th>−0.8</th><td>0.21186</td><td>0.20897</td><td>0.20611</td><td>0.20327</td><td>0.20045</td><td>0.19766</td><td>0.19489</td><td>0.19215</td><td>0.18943</td><td>0.18673</td></tr><tr><th>−0.7</th><td>0.24196</td><td>0.23885</td><td>0.23576</td><td>0.23270</td><td>0.22965</td><td>0.22663</td><td>0.22363</td><td>0.22065</td><td>0.21770</td><td>0.21476</td></tr><tr><th>−0.6</th><td>0.27425</td><td>0.27093</td><td>0.26763</td><td>0.26435</td><td>0.26109</td><td>0.25785</td><td>0.25463</td><td>0.25143</td><td>0.24825</td><td>0.24510</td></tr><tr><th>−0.5</th><td>0.30854</td><td>0.30503</td><td>0.30153</td><td>0.29806</td><td>0.29460</td><td>0.29116</td><td>0.28774</td><td>0.28434</td><td>0.28096</td><td>0.27760</td></tr><tr><td colspan="11"></td></tr><tr><th>−0.4</th><td>0.34458</td><td>0.34090</td><td>0.33724</td><td>0.33360</td><td>0.32997</td><td>0.32636</td><td>0.32276</td><td>0.31918</td><td>0.31561</td><td>0.31207</td></tr><tr><th>−0.3</th><td>0.38209</td><td>0.37828</td><td>0.37448</td><td>0.37070</td><td>0.36693</td><td>0.36317</td><td>0.35942</td><td>0.35569</td><td>0.35197</td><td>0.34827</td></tr><tr><th>−0.2</th><td>0.42074</td><td>0.41683</td><td>0.41294</td><td>0.40905</td><td>0.40517</td><td>0.40129</td><td>0.39743</td><td>0.39358</td><td>0.38974</td><td>0.38591</td></tr><tr><th>−0.1</th><td>0.46017</td><td>0.45620</td><td>0.45224</td><td>0.44828</td><td>0.44433</td><td>0.44038</td><td>0.43644</td><td>0.43251</td><td>0.42858</td><td>0.42465</td></tr><tr><th>−0.0</th><td>0.50000</td><td>0.49601</td><td>0.49202</td><td>0.48803</td><td>0.48405</td><td>0.48006</td><td>0.47608</td><td>0.47210</td><td>0.46812</td><td>0.46414</td></tr><tr><th><i>z</i></th><th>−0.00</th><th>−0.01</th><th>−0.02</th><th>−0.03</th><th>−0.04</th><th>−0.05</th><th>−0.06</th><th>−0.07</th><th>−0.08</th><th>−0.09</th></tr></tbody></table> <table><tbody><tr><th><i>z</i></th><th>+ 0.00</th><th>+ 0.01</th><th>+ 0.02</th><th>+ 0.03</th><th>+ 0.04</th><th>+ 0.05</th><th>+ 0.06</th><th>+ 0.07</th><th>+ 0.08</th><th>+ 0.09</th></tr><tr><th>0.0</th><td>0.50000</td><td>0.50399</td><td>0.50798</td><td>0.51197</td><td>0.51595</td><td>0.51994</td><td>0.52392</td><td>0.52790</td><td>0.53188</td><td>0.53586</td></tr><tr><th>0.1</th><td>0.53983</td><td>0.54380</td><td>0.54776</td><td>0.55172</td><td>0.55567</td><td>0.55962</td><td>0.56360</td><td>0.56749</td><td>0.57142</td><td>0.57535</td></tr><tr><th>0.2</th><td>0.57926</td><td>0.58317</td><td>0.58706</td><td>0.59095</td><td>0.59483</td><td>0.59871</td><td>0.60257</td><td>0.60642</td><td>0.61026</td><td>0.61409</td></tr><tr><th>0.3</th><td>0.61791</td><td>0.62172</td><td>0.62552</td><td>0.62930</td><td>0.63307</td><td>0.63683</td><td>0.64058</td><td>0.64431</td><td>0.64803</td><td>0.65173</td></tr><tr><th>0.4</th><td>0.65542</td><td>0.65910</td><td>0.66276</td><td>0.66640</td><td>0.67003</td><td>0.67364</td><td>0.67724</td><td>0.68082</td><td>0.68439</td><td>0.68793</td></tr><tr><td colspan="1"></td></tr><tr><th>0.5</th><td>0.69146</td><td>0.69497</td><td>0.69847</td><td>0.70194</td><td>0.70540</td><td>0.70884</td><td>0.71226</td><td>0.71566</td><td>0.71904</td><td>0.72240</td></tr><tr><th>0.6</th><td>0.72575</td><td>0.72907</td><td>0.73237</td><td>0.73565</td><td>0.73891</td><td>0.74215</td><td>0.74537</td><td>0.74857</td><td>0.75175</td><td>0.75490</td></tr><tr><th>0.7</th><td>0.75804</td><td>0.76115</td><td>0.76424</td><td>0.76730</td><td>0.77035</td><td>0.77337</td><td>0.77637</td><td>0.77935</td><td>0.78230</td><td>0.78524</td></tr><tr><th>0.8</th><td>0.78814</td><td>0.79103</td><td>0.79389</td><td>0.79673</td><td>0.79955</td><td>0.80234</td><td>0.80511</td><td>0.80785</td><td>0.81057</td><td>0.81327</td></tr><tr><th>0.9</th><td>0.81594</td><td>0.81859</td><td>0.82121</td><td>0.82381</td><td>0.82639</td><td>0.82894</td><td>0.83147</td><td>0.83398</td><td>0.83646</td><td>0.83891</td></tr><tr><td colspan="1"></td></tr><tr><th>1.0</th><td>0.84134</td><td>0.84375</td><td>0.84614</td><td>0.84849</td><td>0.85083</td><td>0.85314</td><td>0.85543</td><td>0.85769</td><td>0.85993</td><td>0.86214</td></tr><tr><th>1.1</th><td>0.86433</td><td>0.86650</td><td>0.86864</td><td>0.87076</td><td>0.87286</td><td>0.87493</td><td>0.87698</td><td>0.87900</td><td>0.88100</td><td>0.88298</td></tr><tr><th>1.2</th><td>0.88493</td><td>0.88686</td><td>0.88877</td><td>0.89065</td><td>0.89251</td><td>0.89435</td><td>0.89617</td><td>0.89796</td><td>0.89973</td><td>0.90147</td></tr><tr><th>1.3</th><td>0.90320</td><td>0.90490</td><td>0.90658</td><td>0.90824</td><td>0.90988</td><td>0.91149</td><td>0.91308</td><td>0.91466</td><td>0.91621</td><td>0.91774</td></tr><tr><th>1.4</th><td>0.91924</td><td>0.92073</td><td>0.92220</td><td>0.92364</td><td>0.92507</td><td>0.92647</td><td>0.92785</td><td>0.92922</td><td>0.93056</td><td>0.93189</td></tr><tr><td colspan="1"></td></tr><tr><th>1.5</th><td>0.93319</td><td>0.93448</td><td>0.93574</td><td>0.93699</td><td>0.93822</td><td>0.93943</td><td>0.94062</td><td>0.94179</td><td>0.94295</td><td>0.94408</td></tr><tr><th>1.6</th><td>0.94520</td><td>0.94630</td><td>0.94738</td><td>0.94845</td><td>0.94950</td><td>0.95053</td><td>0.95154</td><td>0.95254</td><td>0.95352</td><td>0.95449</td></tr><tr><th>1.7</th><td>0.95543</td><td>0.95637</td><td>0.95728</td><td>0.95818</td><td>0.95907</td><td>0.95994</td><td>0.96080</td><td>0.96164</td><td>0.96246</td><td>0.96327</td></tr><tr><th>1.8</th><td>0.96407</td><td>0.96485</td><td>0.96562</td><td>0.96638</td><td>0.96712</td><td>0.96784</td><td>0.96856</td><td>0.96926</td><td>0.96995</td><td>0.97062</td></tr><tr><th>1.9</th><td>0.97128</td><td>0.97193</td><td>0.97257</td><td>0.97320</td><td>0.97381</td><td>0.97441</td><td>0.97500</td><td>0.97558</td><td>0.97615</td><td>0.97670</td></tr><tr><td colspan="1"></td></tr><tr><th>2.0</th><td>0.97725</td><td>0.97778</td><td>0.97831</td><td>0.97882</td><td>0.97932</td><td>0.97982</td><td>0.98030</td><td>0.98077</td><td>0.98124</td><td>0.98169</td></tr><tr><th>2.1</th><td>0.98214</td><td>0.98257</td><td>0.98300</td><td>0.98341</td><td>0.98382</td><td>0.98422</td><td>0.98461</td><td>0.98500</td><td>0.98537</td><td>0.98574</td></tr><tr><th>2.2</th><td>0.98610</td><td>0.98645</td><td>0.98679</td><td>0.98713</td><td>0.98745</td><td>0.98778</td><td>0.98809</td><td>0.98840</td><td>0.98870</td><td>0.98899</td></tr><tr><th>2.3</th><td>0.98928</td><td>0.98956</td><td>0.98983</td><td>0.99010</td><td>0.99036</td><td>0.99061</td><td>0.99086</td><td>0.99111</td><td>0.99134</td><td>0.99158</td></tr><tr><th>2.4</th><td>0.99180</td><td>0.99202</td><td>0.99224</td><td>0.99245</td><td>0.99266</td><td>0.99286</td><td>0.99305</td><td>0.99324</td><td>0.99343</td><td>0.99361</td></tr><tr><td colspan="1"></td></tr><tr><th>2.5</th><td>0.99379</td><td>0.99396</td><td>0.99413</td><td>0.99430</td><td>0.99446</td><td>0.99461</td><td>0.99477</td><td>0.99492</td><td>0.99506</td><td>0.99520</td></tr><tr><th>2.6</th><td>0.99534</td><td>0.99547</td><td>0.99560</td><td>0.99573</td><td>0.99585</td><td>0.99598</td><td>0.99609</td><td>0.99621</td><td>0.99632</td><td>0.99643</td></tr><tr><th>2.7</th><td>0.99653</td><td>0.99664</td><td>0.99674</td><td>0.99683</td><td>0.99693</td><td>0.99702</td><td>0.99711</td><td>0.99720</td><td>0.99728</td><td>0.99736</td></tr><tr><th>2.8</th><td>0.99744</td><td>0.99752</td><td>0.99760</td><td>0.99767</td><td>0.99774</td><td>0.99781</td><td>0.99788</td><td>0.99795</td><td>0.99801</td><td>0.99807</td></tr><tr><th>2.9</th><td>0.99813</td><td>0.99819</td><td>0.99825</td><td>0.99831</td><td>0.99836</td><td>0.99841</td><td>0.99846</td><td>0.99851</td><td>0.99856</td><td>0.99861</td></tr><tr><th>3.0</th><td>0.99865</td><td>0.99869</td><td>0.99874</td><td>0.99878</td><td>0.99882</td><td>0.99886</td><td>0.99889</td><td>0.99893</td><td>0.99896</td><td>0.99900</td></tr><tr><th>3.1</th><td>0.99903</td><td>0.99906</td><td>0.99910</td><td>0.99913</td><td>0.99916</td><td>0.99918</td><td>0.99921</td><td>0.99924</td><td>0.99926</td><td>0.99929</td></tr><tr><th>3.2</th><td>0.99931</td><td>0.99934</td><td>0.99936</td><td>0.99938</td><td>0.99940</td><td>0.99942</td><td>0.99944</td><td>0.99946</td><td>0.99948</td><td>0.99950</td></tr><tr><th>3.3</th><td>0.99952</td><td>0.99953</td><td>0.99955</td><td>0.99957</td><td>0.99958</td><td>0.99960</td><td>0.99961</td><td>0.99962</td><td>0.99964</td><td>0.99965</td></tr><tr><th>3.4</th><td>0.99966</td><td>0.99968</td><td>0.99969</td><td>0.99970</td><td>0.99971</td><td>0.99972</td><td>0.99973</td><td>0.99974</td><td>0.99975</td><td>0.99976</td></tr><tr><th>3.5</th><td>0.99977</td><td>0.99978</td><td>0.99978</td><td>0.99979</td><td>0.99980</td><td>0.99981</td><td>0.99981</td><td>0.99982</td><td>0.99983</td><td>0.99983</td></tr><tr><th>3.6</th><td>0.99984</td><td>0.99985</td><td>0.99985</td><td>0.99986</td><td>0.99986</td><td>0.99987</td><td>0.99987</td><td>0.99988</td><td>0.99988</td><td>0.99989</td></tr><tr><th>3.7</th><td>0.99989</td><td>0.99990</td><td>0.99990</td><td>0.99990</td><td>0.99991</td><td>0.99991</td><td>0.99992</td><td>0.99992</td><td>0.99992</td><td>0.99992</td></tr><tr><th>3.8</th><td>0.99993</td><td>0.99993</td><td>0.99993</td><td>0.99994</td><td>0.99994</td><td>0.99994</td><td>0.99994</td><td>0.99995</td><td>0.99995</td><td>0.99995</td></tr><tr><th>3.9</th><td>0.99995</td><td>0.99995</td><td>0.99996</td><td>0.99996</td><td>0.99996</td><td>0.99996</td><td>0.99996</td><td>0.99996</td><td>0.99997</td><td>0.99997</td></tr><tr><td colspan="1"></td></tr><tr><th><i>z</i></th><th>+0.00</th><th>+0.01</th><th>+0.02</th><th>+0.03</th><th>+0.04</th><th>+0.05</th><th>+0.06</th><th>+0.07</th><th>+0.08</th><th>+0.09</th></tr></tbody></table> <p><a class="footnote-ref" id="fnref:4" href="#fn:4"><sup>4</sup></a> </p> <h3>Complementary cumulative</h3> <p>This table gives a probability that a statistic is greater than Z. : f ( z ) = 1 − Φ ( z ) {\displaystyle f(z)=1-\Phi (z)} </p> <table><tbody><tr><th><i>z</i></th><th>+0.00</th><th>+0.01</th><th>+0.02</th><th>+0.03</th><th>+0.04</th><th rowspan="50"></th><th>+0.05</th><th>+0.06</th><th>+0.07</th><th>+0.08</th><th>+0.09</th></tr><tr><th>0.0</th><td>0.50000</td><td>0.49601</td><td>0.49202</td><td>0.48803</td><td>0.48405</td><td>0.48006</td><td>0.47608</td><td>0.47210</td><td>0.46812</td><td>0.46414</td></tr><tr><th>0.1</th><td>0.46017</td><td>0.45620</td><td>0.45224</td><td>0.44828</td><td>0.44433</td><td>0.44038</td><td>0.43640</td><td>0.43251</td><td>0.42858</td><td>0.42465</td></tr><tr><th>0.2</th><td>0.42074</td><td>0.41683</td><td>0.41294</td><td>0.40905</td><td>0.40517</td><td>0.40129</td><td>0.39743</td><td>0.39358</td><td>0.38974</td><td>0.38591</td></tr><tr><th>0.3</th><td>0.38209</td><td>0.37828</td><td>0.37448</td><td>0.37070</td><td>0.36693</td><td>0.36317</td><td>0.35942</td><td>0.35569</td><td>0.35197</td><td>0.34827</td></tr><tr><th>0.4</th><td>0.34458</td><td>0.34090</td><td>0.33724</td><td>0.33360</td><td>0.32997</td><td>0.32636</td><td>0.32276</td><td>0.31918</td><td>0.31561</td><td>0.31207</td></tr><tr><td colspan="11"></td></tr><tr><th>0.5</th><td>0.30854</td><td>0.30503</td><td>0.30153</td><td>0.29806</td><td>0.29460</td><td>0.29116</td><td>0.28774</td><td>0.28434</td><td>0.28096</td><td>0.27760</td></tr><tr><th>0.6</th><td>0.27425</td><td>0.27093</td><td>0.26763</td><td>0.26435</td><td>0.26109</td><td>0.25785</td><td>0.25463</td><td>0.25143</td><td>0.24825</td><td>0.24510</td></tr><tr><th>0.7</th><td>0.24196</td><td>0.23885</td><td>0.23576</td><td>0.23270</td><td>0.22965</td><td>0.22663</td><td>0.22363</td><td>0.22065</td><td>0.21770</td><td>0.21476</td></tr><tr><th>0.8</th><td>0.21186</td><td>0.20897</td><td>0.20611</td><td>0.20327</td><td>0.20045</td><td>0.19766</td><td>0.19489</td><td>0.19215</td><td>0.18943</td><td>0.18673</td></tr><tr><th>0.9</th><td>0.18406</td><td>0.18141</td><td>0.17879</td><td>0.17619</td><td>0.17361</td><td>0.17106</td><td>0.16853</td><td>0.16602</td><td>0.16354</td><td>0.16109</td></tr><tr><td colspan="11"></td></tr><tr><th>1.0</th><td>0.15866</td><td>0.15625</td><td>0.15386</td><td>0.15151</td><td>0.14917</td><td>0.14686</td><td>0.14457</td><td>0.14231</td><td>0.14007</td><td>0.13786</td></tr><tr><th>1.1</th><td>0.13567</td><td>0.13350</td><td>0.13136</td><td>0.12924</td><td>0.12714</td><td>0.12507</td><td>0.12302</td><td>0.12100</td><td>0.11900</td><td>0.11702</td></tr><tr><th>1.2</th><td>0.11507</td><td>0.11314</td><td>0.11123</td><td>0.10935</td><td>0.10749</td><td>0.10565</td><td>0.10383</td><td>0.10204</td><td>0.10027</td><td>0.09853</td></tr><tr><th>1.3</th><td>0.09680</td><td>0.09510</td><td>0.09342</td><td>0.09176</td><td>0.09012</td><td>0.08851</td><td>0.08692</td><td>0.08534</td><td>0.08379</td><td>0.08226</td></tr><tr><th>1.4</th><td>0.08076</td><td>0.07927</td><td>0.07780</td><td>0.07636</td><td>0.07493</td><td>0.07353</td><td>0.07215</td><td>0.07078</td><td>0.06944</td><td>0.06811</td></tr><tr><td colspan="11"></td></tr><tr><th>1.5</th><td>0.06681</td><td>0.06552</td><td>0.06426</td><td>0.06301</td><td>0.06178</td><td>0.06057</td><td>0.05938</td><td>0.05821</td><td>0.05705</td><td>0.05592</td></tr><tr><th>1.6</th><td>0.05480</td><td>0.05370</td><td>0.05262</td><td>0.05155</td><td>0.05050</td><td>0.04947</td><td>0.04846</td><td>0.04746</td><td>0.04648</td><td>0.04551</td></tr><tr><th>1.7</th><td>0.04457</td><td>0.04363</td><td>0.04272</td><td>0.04182</td><td>0.04093</td><td>0.04006</td><td>0.03920</td><td>0.03836</td><td>0.03754</td><td>0.03673</td></tr><tr><th>1.8</th><td>0.03593</td><td>0.03515</td><td>0.03438</td><td>0.03362</td><td>0.03288</td><td>0.03216</td><td>0.03144</td><td>0.03074</td><td>0.03005</td><td>0.02938</td></tr><tr><th>1.9</th><td>0.02872</td><td>0.02807</td><td>0.02743</td><td>0.02680</td><td>0.02619</td><td>0.02559</td><td>0.02500</td><td>0.02442</td><td>0.02385</td><td>0.02330</td></tr><tr><td colspan="11"></td></tr><tr><th>2.0</th><td>0.02275</td><td>0.02222</td><td>0.02169</td><td>0.02118</td><td>0.02068</td><td>0.02018</td><td>0.01970</td><td>0.01923</td><td>0.01876</td><td>0.01831</td></tr><tr><th>2.1</th><td>0.01786</td><td>0.01743</td><td>0.01700</td><td>0.01659</td><td>0.01618</td><td>0.01578</td><td>0.01539</td><td>0.01500</td><td>0.01463</td><td>0.01426</td></tr><tr><th>2.2</th><td>0.01390</td><td>0.01355</td><td>0.01321</td><td>0.01287</td><td>0.01255</td><td>0.01222</td><td>0.01191</td><td>0.01160</td><td>0.01130</td><td>0.01101</td></tr><tr><th>2.3</th><td>0.01072</td><td>0.01044</td><td>0.01017</td><td>0.00990</td><td>0.00964</td><td>0.00939</td><td>0.00914</td><td>0.00889</td><td>0.00866</td><td>0.00842</td></tr><tr><th>2.4</th><td>0.00820</td><td>0.00798</td><td>0.00776</td><td>0.00755</td><td>0.00734</td><td>0.00714</td><td>0.00695</td><td>0.00676</td><td>0.00657</td><td>0.00639</td></tr><tr><td colspan="11"></td></tr><tr><th>2.5</th><td>0.00621</td><td>0.00604</td><td>0.00587</td><td>0.00570</td><td>0.00554</td><td>0.00539</td><td>0.00523</td><td>0.00508</td><td>0.00494</td><td>0.00480</td></tr><tr><th>2.6</th><td>0.00466</td><td>0.00453</td><td>0.00440</td><td>0.00427</td><td>0.00415</td><td>0.00402</td><td>0.00391</td><td>0.00379</td><td>0.00368</td><td>0.00357</td></tr><tr><th>2.7</th><td>0.00347</td><td>0.00336</td><td>0.00326</td><td>0.00317</td><td>0.00307</td><td>0.00298</td><td>0.00289</td><td>0.00280</td><td>0.00272</td><td>0.00264</td></tr><tr><th>2.8</th><td>0.00256</td><td>0.00248</td><td>0.00240</td><td>0.00233</td><td>0.00226</td><td>0.00219</td><td>0.00212</td><td>0.00205</td><td>0.00199</td><td>0.00193</td></tr><tr><th>2.9</th><td>0.00187</td><td>0.00181</td><td>0.00175</td><td>0.00169</td><td>0.00164</td><td>0.00159</td><td>0.00154</td><td>0.00149</td><td>0.00144</td><td>0.00139</td></tr><tr><td colspan="11"></td></tr><tr><th>3.0</th><td>0.00135</td><td>0.00131</td><td>0.00126</td><td>0.00122</td><td>0.00118</td><td>0.00114</td><td>0.00111</td><td>0.00107</td><td>0.00104</td><td>0.00100</td></tr><tr><th>3.1</th><td>0.00097</td><td>0.00094</td><td>0.00090</td><td>0.00087</td><td>0.00084</td><td>0.00082</td><td>0.00079</td><td>0.00076</td><td>0.00074</td><td>0.00071</td></tr><tr><th>3.2</th><td>0.00069</td><td>0.00066</td><td>0.00064</td><td>0.00062</td><td>0.00060</td><td>0.00058</td><td>0.00056</td><td>0.00054</td><td>0.00052</td><td>0.00050</td></tr><tr><th>3.3</th><td>0.00048</td><td>0.00047</td><td>0.00045</td><td>0.00043</td><td>0.00042</td><td>0.00040</td><td>0.00039</td><td>0.00038</td><td>0.00036</td><td>0.00035</td></tr><tr><th>3.4</th><td>0.00034</td><td>0.00032</td><td>0.00031</td><td>0.00030</td><td>0.00029</td><td>0.00028</td><td>0.00027</td><td>0.00026</td><td>0.00025</td><td>0.00024</td></tr><tr><td colspan="11"></td></tr><tr><th>3.5</th><td>0.00023</td><td>0.00022</td><td>0.00022</td><td>0.00021</td><td>0.00020</td><td>0.00019</td><td>0.00019</td><td>0.00018</td><td>0.00017</td><td>0.00017</td></tr><tr><th>3.6</th><td>0.00016</td><td>0.00015</td><td>0.00015</td><td>0.00014</td><td>0.00014</td><td>0.00013</td><td>0.00013</td><td>0.00012</td><td>0.00012</td><td>0.00011</td></tr><tr><th>3.7</th><td>0.00011</td><td>0.00010</td><td>0.00010</td><td>0.00010</td><td>0.00009</td><td>0.00009</td><td>0.00008</td><td>0.00008</td><td>0.00008</td><td>0.00008</td></tr><tr><th>3.8</th><td>0.00007</td><td>0.00007</td><td>0.00007</td><td>0.00006</td><td>0.00006</td><td>0.00006</td><td>0.00006</td><td>0.00005</td><td>0.00005</td><td>0.00005</td></tr><tr><th>3.9</th><td>0.00005</td><td>0.00005</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00004</td><td>0.00003</td><td>0.00003</td></tr><tr><td colspan="11"></td></tr><tr><th>4.0</th><td>0.00003</td><td>0.00003</td><td>0.00003</td><td>0.00003</td><td>0.00003</td><td>0.00003</td><td>0.00002</td><td>0.00002</td><td>0.00002</td><td>0.00002</td></tr></tbody></table> <p><a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a> This table gives a probability that a statistic is greater than Z, for large integer Z values. </p> <table><tbody><tr><th><i>z</i></th><th>+0</th><th>+1</th><th>+2</th><th>+3</th><th>+4</th><th rowspan="38"></th><th>+5</th><th>+6</th><th>+7</th><th>+8</th><th>+9</th></tr><tr><th>0</th><td>5.00000×10−1</td><td>1.58655×10−1</td><td>2.27501×10−2</td><td>1.34990×10−3</td><td>3.16712×10−5</td><td>2.86652×10−7</td><td>9.86588×10−10</td><td>1.27981×10−12</td><td>6.22096×10−16</td><td>1.12859×10−19</td></tr><tr><th>10</th><td>7.61985×10−24</td><td>1.91066×10−28</td><td>1.77648×10−33</td><td>6.11716×10−39</td><td>7.79354×10−45</td><td>3.67097×10−51</td><td>6.38875×10−58</td><td>4.10600×10−65</td><td>9.74095×10−73</td><td>8.52722×10−81</td></tr><tr><th>20</th><td>2.75362×10−89</td><td>3.27928×10−98</td><td>1.43989×10−107</td><td>2.33064×10−117</td><td>1.39039×10−127</td><td>3.05670×10−138</td><td>2.47606×10−149</td><td>7.38948×10−161</td><td>8.12387×10−173</td><td>3.28979×10−185</td></tr><tr><th>30</th><td>4.90671×10−198</td><td>2.69525×10−211</td><td>5.45208×10−225</td><td>4.06119×10−239</td><td>1.11390×10−253</td><td>1.12491×10−268</td><td>4.18262×10−284</td><td>5.72557×10−300</td><td>2.88543×10−316</td><td>5.35312×10−333</td></tr><tr><th>40</th><td>3.65589×10−350</td><td>9.19086×10−368</td><td>8.50515×10−386</td><td>2.89707×10−404</td><td>3.63224×10−423</td><td>1.67618×10−442</td><td>2.84699×10−462</td><td>1.77976×10−482</td><td>4.09484×10−503</td><td>3.46743×10−524</td></tr><tr><th>50</th><td>1.08060×10−545</td><td>1.23937×10−567</td><td>5.23127×10−590</td><td>8.12606×10−613</td><td>4.64529×10−636</td><td>9.77237×10−660</td><td>7.56547×10−684</td><td>2.15534×10−708</td><td>2.25962×10−733</td><td>8.71741×10−759</td></tr><tr><th>60</th><td>1.23757×10−784</td><td>6.46517×10−811</td><td>1.24283×10−837</td><td>8.79146×10−865</td><td>2.28836×10−892</td><td>2.19180×10−920</td><td>7.72476×10−949</td><td>1.00178×10−977</td><td>4.78041×10−1007</td><td>8.39374×10−1037</td></tr><tr><th>70</th><td>5.42304×10−1067</td><td>1.28921×10−1097</td><td>1.12771×10−1128</td><td>3.62960×10−1160</td><td>4.29841×10−1192</td><td>1.87302×10−1224</td><td>3.00302×10−1257</td><td>1.77155×10−1290</td><td>3.84530×10−1324</td><td>3.07102×10−1358</td></tr></tbody></table> <h2 id="examples-of-use">Examples of use</h2> <p>A professor's exam scores are approximately distributed normally with mean 80 and standard deviation 5. Only a <i>cumulative from mean</i> table is available. </p> <ul><li>What is the probability that a student scores an 82 or less? P ( X ≤ 82 ) = P ( Z ≤ 82 − 80 5 ) = P ( Z ≤ 0.40 ) = 0.15542 + 0.5 = 0.65542 {\displaystyle {\begin{aligned}P(X\leq 82)&=P\!\!\left(Z\leq {\frac {82-80}{5}}\right)\\&=P(Z\leq 0.40)\\[2pt]&=0.15542+0.5\\[2pt]&=0.65542\end{aligned}}} </li> <li>What is the probability that a student scores a 90 or more? P ( X ≥ 90 ) = P ( Z ≥ 90 − 80 5 ) = P ( Z ≥ 2.00 ) = 1 − P ( Z ≤ 2.00 ) = 1 − ( 0.47725 + 0.5 ) = 0.02275 {\displaystyle {\begin{aligned}P(X\geq 90)&=P\!\!\left(Z\geq {\frac {90-80}{5}}\right)\\&=P(Z\geq 2.00)\\[2pt]&=1-P(Z\leq 2.00)\\[2pt]&=1-(0.47725+0.5)\\[2pt]&=0.02275\end{aligned}}} </li> <li>What is the probability that a student scores a 74 or less? P ( X ≤ 74 ) = P ( Z ≤ 74 − 80 5 ) = P ( Z ≤ − 1.20 ) {\displaystyle {\begin{aligned}P(X\leq 74)&=P\!\!\left(Z\leq {\frac {74-80}{5}}\right)\\&=P(Z\leq -1.20)\end{aligned}}} Since this table does not include negatives, the process involves the following additional step: = P ( Z ≥ 1.20 ) = 1 − ( 0.38493 + 0.5 ) = 0.11507 {\displaystyle {\begin{aligned}\qquad \qquad \quad ={}&P(Z\geq 1.20)\\[2pt]={}&1-(0.38493+0.5)\\[2pt]={}&0.11507\end{aligned}}} </li> <li>What is the probability that a student scores between 74 and 82? P ( 74 ≤ X ≤ 82 ) = P ( X ≤ 82 ) − P ( X ≤ 74 ) = 0.65542 − 0.11507 = 0.54035 {\displaystyle {\begin{aligned}P(74\leq X\leq 82)&=P(X\leq 82)-P(X\leq 74)\\[2pt]&=0.65542-0.11507\\[2pt]&=0.54035\end{aligned}}} </li> <li>What is the probability that an average of three scores is 82 or less? P ( X ≤ 82 ) = P ( Z ≤ 82 − 80 5 / 3 ) = P ( Z ≤ 0.69 ) = 0.2549 + 0.5 = 0.7549 {\displaystyle {\begin{aligned}P(X\leq 82)&=P\left(Z\leq {\frac {82-80}{5/{\sqrt {3}}}}\right)\\&=P(Z\leq 0.69)\\[2pt]&=0.2549+0.5\\[2pt]&=0.7549\end{aligned}}} </li></ul> <h2 id="see-also">See also</h2> <ul> <li>Mathematics portal</li></ul> <ul><li><a href="/facts/68%25E2%2580%259395%25E2%2580%259399.7_rule/IDaupJFh">68–95–99.7 rule</a></li> <li><a href="/facts/Student%2527s_t-distribution/DeT1SDqH"><i>t</i>-distribution table</a></li></ul> <h2 id="references">References</h2> <ol> <li id="fn:1"><p>"Z Table. History of Z Table. Z Score". Retrieved 21 December 2018. <a href="https://www.ztable.net/" target="_blank">https://www.ztable.net/</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li> <li id="fn:2"><p>Larson, Ron; Farber, Elizabeth (2004). Elementary Statistics: Picturing the World. 清华大学出版社. p. 214. ISBN 7-302-09723-2. <a href="7-302-09723-2" target="_blank">7-302-09723-2</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li> <li id="fn:3"><p>"How to use a Z Table". ztable.io. Retrieved 9 January 2023. <a href="https://ztable.io/" target="_blank">https://ztable.io/</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li> <li id="fn:4"><p>0.5 + each value in Cumulative from mean table <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li> <li id="fn:5"><p>0.5 − each value in Cumulative from mean (0 to Z) table <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li> </ol>