1007:
523:
167.8 175.4 176.1 166 174.7 170.2 178.9 180.4 174.6 174.5 182.4 173.4 167.4 170.7 180.6 169.6 176.2 176.3 175.1 178.7 167.2 180.2 180.3 164.7 167.9 179.6 164.9 173.2 180.3 168 175.5 172.9 182.2 166.7 172.4 181.9 175.9 176.8 179.6 166 171.5 180.6 175.5 173.2 178.8 168.3 170.3 174.2 168 172.6 163.3
515:
Where there are a large number of samples a quick reasonable estimate of the mean and standard deviation can be got by grouping the samples into classes using equal size ranges. This introduces a quantization error but is normally accurate enough for most purposes if 10 or more classes are used.
105:
The method depends on estimating the mean and rounding to an easy value to calculate with. This value is then subtracted from all the sample values. When the samples are classed into equal size ranges a central class is chosen and the count of ranges from that is used in the calculations. For
524:
172.5 163.4 165.9 178.2 174.6 174.3 170.5 169.7 176.2 175.1 177 173.5 173.6 174.3 174.4 171.1 173.3 164.6 173 177.9 166.5 159.6 170.5 174.7 182 172.7 175.9 171.5 167.1 176.9 181.7 170.7 177.5 170.9 178.1 174.3 173.3 169.2 178.2 179.4 187.6 186.4 178.1 174 177.1 163.3 178.1 179.1 175.6
528:
The minimum and maximum are 159.6 and 187.6 we can group them as follows rounding the numbers down. The class size (CS) is 3. The assumed mean is the centre of the range from 174 to 177 which is 175.5. The differences are counted in classes.
913:
993:
505:
433:
288:
376:
227:
43:
of a data set. It simplifies calculating accurate values by hand. Its interest today is chiefly historical but it can be used to quickly estimate these statistics. There are other
171:
92:
of these 15 deviations from the assumed mean is therefore −30/15 = −2. Therefore, that is what we need to add to the assumed mean to get the correct mean:
323:
834:
63:
Suppose we start with a plausible initial guess that the mean is about 240. Then the deviations from this "assumed" mean are the following:
927:
448:
384:
233:
334:
177:
439:
122:
67:−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22
294:
1006:
47:
which are more suited for computers which also ensure more accurate results than the obvious methods.
1012:
44:
40:
36:
1028:
1022:
17:
106:
example, for people's heights a value of 1.75m might be used as the assumed mean.
1002:
28:
908:{\displaystyle x_{0}+CS\times {\frac {A}{N}}=175.5+3\times -55/100=173.85}
59:
219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262
988:{\displaystyle CS{\sqrt {\frac {B-{\frac {A^{2}}{N}}}{N-1}}}=5.57}
88:
and so on. We are left with a sum of −30. The
500:{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N-1}}}\,}
428:{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N}}}\,}
930:
837:
451:
387:
337:
297:
236:
180:
125:
55:
First: The mean of the following numbers is sought:
918:which is very close to the actual mean of 173.846.
531:
987:
907:
499:
427:
370:
317:
282:
221:
165:
78:15 and −17 almost cancel, leaving −2,
8:
283:{\displaystyle B=\sum _{i=1}^{N}d_{i}^{2}\,}
75:22 and −21 almost cancel, leaving +1,
953:
947:
937:
929:
891:
860:
842:
836:
496:
475:
458:
450:
438:or for a sample standard deviation using
424:
411:
394:
386:
371:{\displaystyle {\overline {x}}=x_{0}+D\,}
367:
355:
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314:
304:
296:
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218:
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202:
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130:
124:
222:{\displaystyle A=\sum _{i=1}^{N}d_{i}\,}
921:The standard deviation is estimated as
7:
71:In adding these up, one finds that:
166:{\displaystyle d_{i}=x_{i}-x_{0}\,}
96:correct mean = 240 − 2 = 238.
519:For instance with the exception,
318:{\displaystyle D={\frac {A}{N}}\,}
25:
828:The mean is then estimated to be
109:For a data set with assumed mean
84:7 + 4 cancels −6 − 5,
1005:
35:is a method for calculating the
1:
343:
533:Observed numbers in ranges
1045:
511:Example using class ranges
45:rapid calculation methods
989:
909:
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429:
372:
319:
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81:9 and −9 cancel,
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18:Assumed mean formula
534:
440:Bessel's correction
278:
1013:Mathematics portal
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905:
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41:standard deviation
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16:(Redirected from
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37:arithmetic mean
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54:
33:assumed mean
32:
26:
541:tally-count
999:References
553:freq×diff
547:class diff
29:statistics
970:−
945:−
886:−
883:×
858:×
550:freq×diff
544:frequency
487:−
466:−
453:σ
402:−
389:σ
344:¯
245:∑
189:∑
150:−
116:suppose:
1023:Category
822:B = 371
789:186—188
770:183—185
744:180—182
715:177—179
681:174—176
652:171—173
626:168—170
601:165—167
578:162—164
558:159—161
819:A = −55
814:N = 100
90:average
51:Example
903:173.85
101:Method
31:, the
1029:Means
874:175.5
538:Range
328:Then
983:5.57
809:Sum
751:////
748:////
725:////
722:////
719:////
697:////
694:////
691:////
688:////
685:////
662:////
659:////
656:////
635:///
633:////
630:////
608:////
605:////
582:////
39:and
897:100
804:32
792://
765:44
739:16
676:16
673:−16
647:52
644:−26
621:90
618:−30
596:96
593:−24
573:25
27:In
1025::
889:55
784:0
762:22
756:11
753:/
736:16
730:16
727:/
710:0
701:25
670:−1
667:16
664:/
641:−2
638:13
615:−3
612:10
590:−4
584:/
570:−5
567:−5
561:/
442::
980:=
973:1
967:N
960:N
955:2
951:A
942:B
935:S
932:C
900:=
893:/
880:3
877:+
871:=
866:N
863:A
855:S
852:C
849:+
844:0
840:x
801:8
798:4
795:2
781:0
778:3
775:0
759:2
733:1
707:0
704:0
587:6
564:1
490:1
484:N
477:2
473:D
469:N
463:B
456:=
419:N
413:2
409:D
405:N
399:B
392:=
365:D
362:+
357:0
353:x
349:=
341:x
310:N
307:A
302:=
299:D
275:2
270:i
266:d
260:N
255:1
252:=
249:i
241:=
238:B
214:i
210:d
204:N
199:1
196:=
193:i
185:=
182:A
158:0
154:x
145:i
141:x
137:=
132:i
128:d
114:0
111:x
20:)
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