中位數
中位數特性 > 常態分配下的中位數

常態分配下的平均數、中位數、眾數都是同一個位置。目前最為世人熟知的是平均數的抽樣分配會是常態分配，期望值為母體平均數 \mu 且變異數為母體變異數( \sigma _{2})。統計學對常態分配的母體平均數統計量說明甚多，並發展完善。那麼中位數可基於機率分配模擬器和數值分析發展，在n個獨立隨機變數來自常態分配可生成n個隨機樣本，則E(樣本中位數)= \mu 且Var(樣本中位數)= \sigma _{2}k(n)，其中，k(n)受到樣本個數(n)影響。當樣本個數介於2至200時，兩者的關係不明顯，但可計算出樣本個數和k(n)的關聯表。
| n | k(n) | n | k(n) | n | k(n) |
| --- | --- | --- | --- | --- | --- |
| 2 | 0.500267128 | 70 | 0.021985179 | 138 | 0.011271806 |
| 3 | 0.448703237 | 71 | 0.021403637 | 139 | 0.011269587 |
| 4 | 0.298172500 | 72 | 0.021393271 | 140 | 0.011109049 |
| 5 | 0.286770401 | 73 | 0.020840845 | 141 | 0.011111745 |
| 6 | 0.214713620 | 74 | 0.020830427 | 142 | 0.010959968 |
| 7 | 0.210476952 | 75 | 0.020295864 | 143 | 0.010962027 |
| 8 | 0.168172011 | 76 | 0.020294599 | 144 | 0.010810205 |
| 9 | 0.166171644 | 77 | 0.019776971 | 145 | 0.010809127 |
| 10 | 0.138304145 | 78 | 0.019777466 | 146 | 0.010661452 |
| 11 | 0.137221972 | 79 | 0.019291777 | 147 | 0.010659591 |
| 12 | 0.117603985 | 80 | 0.019294767 | 148 | 0.010513172 |
| 13 | 0.116875871 | 81 | 0.018831955 | 149 | 0.010523498 |
| 14 | 0.102209683 | 82 | 0.018826854 | 150 | 0.010377973 |
| 15 | 0.101704592 | 83 | 0.018394657 | 151 | 0.010379735 |
| 16 | 0.090397468 | 84 | 0.018390467 | 152 | 0.010244606 |
| 17 | 0.090046842 | 85 | 0.017972657 | 153 | 0.010247290 |
| 18 | 0.081017991 | 86 | 0.017972309 | 154 | 0.010109136 |
| 19 | 0.080776427 | 87 | 0.017567447 | 155 | 0.010114347 |
| 20 | 0.073450103 | 88 | 0.017564340 | 156 | 0.009986419 |
| 21 | 0.073284584 | 89 | 0.017187295 | 157 | 0.009984465 |
| 22 | 0.067168338 | 90 | 0.017189110 | 158 | 0.009862704 |
| 23 | 0.067002164 | 91 | 0.016812903 | 159 | 0.009858886 |
| 24 | 0.061881619 | 92 | 0.016813666 | 160 | 0.009735345 |
| 25 | 0.061762647 | 93 | 0.016466660 | 161 | 0.009736185 |
| 26 | 0.057309720 | 94 | 0.016462668 | 162 | 0.009617128 |
| 27 | 0.057271174 | 95 | 0.016125488 | 163 | 0.009619325 |
| 28 | 0.053440064 | 96 | 0.016119237 | 164 | 0.009501480 |
| 29 | 0.053332370 | 97 | 0.015802880 | 165 | 0.009502525 |
| 30 | 0.049992614 | 98 | 0.015797856 | 166 | 0.009389839 |
| 31 | 0.049937448 | 99 | 0.015492872 | 167 | 0.009388423 |
| 32 | 0.047029351 | 100 | 0.015490432 | 168 | 0.009279058 |
| 33 | 0.046965211 | 101 | 0.015190773 | 169 | 0.009277712 |
| 34 | 0.044337988 | 102 | 0.015189776 | 170 | 0.009169514 |
| 35 | 0.044336558 | 103 | 0.014904567 | 171 | 0.009169768 |
| 36 | 0.041990927 | 104 | 0.014896640 | 172 | 0.009061071 |
| 37 | 0.041942218 | 105 | 0.014628725 | 173 | 0.009060657 |
| 38 | 0.039852927 | 106 | 0.014623638 | 174 | 0.008961003 |
| 39 | 0.039832458 | 107 | 0.014359452 | 175 | 0.008957769 |
| 40 | 0.037939073 | 108 | 0.014359166 | 176 | 0.008860612 |
| 41 | 0.037904745 | 109 | 0.014100614 | 177 | 0.008859363 |
| 42 | 0.036184274 | 110 | 0.014104129 | 178 | 0.008762802 |
| 43 | 0.036152192 | 111 | 0.013856818 | 179 | 0.008760489 |
| 44 | 0.034579591 | 112 | 0.013854712 | 180 | 0.008665028 |
| 45 | 0.034577569 | 113 | 0.013609600 | 181 | 0.008663662 |
| 46 | 0.033133177 | 114 | 0.013610680 | 182 | 0.008571695 |
| 47 | 0.033118807 | 115 | 0.013383360 | 183 | 0.008570240 |
| 48 | 0.031791145 | 116 | 0.013382329 | 184 | 0.008475410 |
| 49 | 0.031783399 | 117 | 0.013153728 | 185 | 0.008477845 |
| 50 | 0.030548873 | 118 | 0.013156167 | 186 | 0.008388634 |
| 51 | 0.030533811 | 119 | 0.012938560 | 187 | 0.008384818 |
| 52 | 0.029411882 | 120 | 0.012939455 | 188 | 0.008300454 |
| 53 | 0.029402885 | 121 | 0.012729706 | 189 | 0.008300175 |
| 54 | 0.028347691 | 122 | 0.012731381 | 190 | 0.008214157 |
| 55 | 0.028342062 | 123 | 0.012533040 | 191 | 0.008211878 |
| 56 | 0.027348747 | 124 | 0.012525181 | 192 | 0.008130539 |
| 57 | 0.027350473 | 125 | 0.012333899 | 193 | 0.008128310 |
| 58 | 0.026442809 | 126 | 0.012334408 | 194 | 0.008045347 |
| 59 | 0.026436289 | 127 | 0.012141084 | 195 | 0.008041810 |
| 60 | 0.025573242 | 128 | 0.012138522 | 196 | 0.007964784 |
| 61 | 0.025575279 | 129 | 0.011964057 | 197 | 0.007961234 |
| 62 | 0.024780610 | 130 | 0.011961887 | 198 | 0.007882679 |
| 63 | 0.024751923 | 131 | 0.011782874 | 199 | 0.007882009 |
| 64 | 0.024005574 | 132 | 0.011779941 | 200 | 0.007806200 |
| 65 | 0.024006688 | 133 | 0.011604216 | 201 | 0.007801090 |
| 66 | 0.023304209 | 134 | 0.011600908 | 202 | 0.007729016 |
| 67 | 0.023287460 | 135 | 0.011433315 | 203 | 0.007728333 |
| 68 | 0.022616908 | 136 | 0.011438587 | 204 | 0.007654504 |
| 69 | 0.022624425 | 137 | 0.011271806 | 205 | 0.007652196 |
如果樣本個數超過200，但不超過1000時，兩者有明顯的關係，並且受到樣本個數是否為奇數或偶數影響。此時可使用迴歸分析尋找兩者的關係。
1. 樣本個數為偶數，迴歸式為k(n) = 0.0000148965 + 1.5599936862 / n。
2. 樣本個數為奇數，迴歸式為k(n) = 0.0000084608 + 1.5674001064 / n。
由此可得到樣本中位數的變異數和母體常態分配的變異數形成穩定的對應關係。