正态曲线拟合-公卫百科-公共卫生科学数据中心

正态曲线拟合

摘要: 计量资料若频数分布近于正态分布，可拟合正态曲线以估计其总体分布。

计量资料若频数分布近于正态分布，可拟合正态曲线以估计其总体分布。此法常用于重复测定误差的估计和正常值范围的估计。正态曲线拟合的方法步骤如下：

(1)计算均数X和校正标准差sc。分别按式(1)、(2)。

式中f为各组段的频数，i为组距，X₀为假定均数，x=(组中值-X₀)/i，n为样本含量，1/12为归组校正数，或称Sheppard校正数。用频数表计算标准差时，以组中值代替本组段各观察值，而组中值两侧的频数并非对称分布，而是离均数近的一侧稍多，这样算得的标准差往往略微夸大了实际的变异度。采用校正数就是为了消除此夸大的部分，使拟合的正态曲线更接近实际的频数分布。若不用频数表而直接按原观察值计算标准差，则无须校正。

(2)计算标准正态离差u。按式(3)。

式中X为各组段上、下限。如表2资料X=14.18，sc=1.21，第一个组段的下限为10.0，因此，u=|10-14.18|/1.21=3.45，余类推，如表2第(4)栏。在X所在组段，还应写出X=X时，u=0.00。

(3) 求z。由表1查出与u对应的标准正态曲线的纵坐标z，如表2第(5)栏。

(4)计算与原资料频数分布各组段上、下限相应的正态曲线之纵坐标Y。按式(4)。

式中sc/i为以组距为单位的校正标准差。如表2资料n=110，i=0.5，sc=1.21，则常数。因此，表2第(5)栏乘以45.4545即得Y，如表2第(6)栏。比如z=0.001时，Y=0.001(45.4545)=0.05;又u=0.00，z=0.399时，Y=0.399(45.4545)=18.14，即拟合的正态曲线上，X=时的纵坐标，也就是曲线的顶点。

表1 标准正态曲线的纵坐标z

u	.00	.01	.02	.03	.04	.05	.06	.07	.08	.09
.0	.399	.399	.399	.399	.399	.398	.398	.398	.398	.397
.1 .2 .3 .4 .5	.397 .391 .381 .368 .352	.397 .390 .380 .367 .350	.396 .389 .379 .365 .348	.396 .389 .378 .364 .347	.395 .388 .377 .362 .345	.394 .387 .375 .361 .343	.394 .386 .374 .359 .341	.393 .385 .373 .357 .339	.393 .384 .371 .356 .337	.392 .383 .370 .354 .335
.6 .7 .8 .9 1.0	.333 .312 .290 .266 .242	.331 .310 .287 .264 .240	.329 .308 .285 .261 .237	.327 .306 .283 .259 .235	.325 .303 .280 .256 .232	.323 .301 .278 .254 .230	.321 .299 .276 .252 .227	.319 .297 .273 .249 .225	.317 .294 .271 .247 .223	.314 .292 .268 .244 .220
1.1 1.2 1.3 1.4 1.5	.218 .194 .171 .150 .130	.215 .192 .169 .148 .128	.213 .190 .167 .146 .126	.211 .187 .165 .144 .124	.208 .185 .163 .141 .122	.206 .183 .160 .139 .120	.204 .180 .158 .137 .118	.201 .178 .156 .135 .116	.199 .176 .154 .133 .115	.197 .174 .152 .131 .113
1.6 1.7 1.8 1.9 2.0	.111 .094 .079 .066 .054	.109 .092 .078 .064 .053	.107 .091 .076 .063 .052	.106 .089 .075 .062 .051	.104 .088 .073 .061 .050	.102 .086 .072 .060 .049	.101 .085 .071 .058 .048	.099 .083 .069 .057 .047	.097 .082 .068 .056 .046	.096 .080 067 .055 .045
2.1 2.2 2.3 2.4 2.5	.044 .035 .028 .022 .018	.043 .035 .028 .022 .017	.042 .034 .027 .021 .017	.041 .033 .026 .021 .016	.040 .032 .026 .020 .016	.040 .032 .025 .020 .015	.039 .031 .025 .019 .015	.038 .030 .024 .019 .015	.037 .030 .023 .018 .014	.036 .029 .023 .018 .014
2.6 2.7 2.8 2.9 3.0	.014 .010 .008 .006 .004	.013 .010 .008 .006 .004	.013 .010 .007 .006 .004	.013 .010 .007 .005 .004	.012 .009 .007 .005 .004	.012 .009 .007 .005 .004	.012 .009 .007 .005 .004	.011 .009 .006 .005 .004	.011 .008 .006 .005 .003	.011 .008 .006 .005 .003
3.1 3.2 3.3 3.4 3.5	.003 .002 .002 .001 .001	.003 .002 .002 .001 .001	.003 .002 .002 .001 .001	.003 .002 .002 .001 .001	.003 .002 .002 .001 .001	.003 .002 .001 .001 .001	.003 .002 .001 .001 .001	.003 .002 .001 .001 .001	.003 .002 .001 .001 .001	.002 .002 .001 .001 .001
3.6	.001	.001	.001	.001	.001	.001	.000	000	.000	.000