996:
512:
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
504:
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.
94:
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
513:
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
517:
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.
902:
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32:
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
160:
81:
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:
312:
823:
52:
Suppose we start with a plausible initial guess that the mean is about 240. Then the deviations from this "assumed" mean are the following:
916:
437:
373:
222:
33:
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166:
111:
56:−21, −17, −14, −12, −9, −6, −5, −4, 0, 1, 4, 7, 9, 15, 22
283:
36:
which are more suited for computers which also ensure more accurate results than the obvious methods.
428:
8:
1001:
29:
995:
95:
example, for people's heights a value of 1.75m might be used as the assumed mean.
25:
1017:
1011:
897:{\displaystyle x_{0}+CS\times {\frac {A}{N}}=175.5+3\times -55/100=173.85}
48:
219, 223, 226, 228, 231, 234, 235, 236, 240, 241, 244, 247, 249, 255, 262
17:
977:{\displaystyle CS{\sqrt {\frac {B-{\frac {A^{2}}{N}}}{N-1}}}=5.57}
77:
and so on. We are left with a sum of −30. The
489:{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N-1}}}\,}
417:{\displaystyle \sigma ={\sqrt {\frac {B-ND^{2}}{N}}}\,}
919:
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44:
First: The mean of the following numbers is sought:
907:which is very close to the actual mean of 173.846.
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67:15 and −17 almost cancel, leaving −2,
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499:
272:{\displaystyle B=\sum _{i=1}^{N}d_{i}^{2}\,}
64:22 and −21 almost cancel, leaving +1,
485:
427:or for a sample standard deviation using
413:
360:{\displaystyle {\overline {x}}=x_{0}+D\,}
356:
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268:
207:
151:
211:{\displaystyle A=\sum _{i=1}^{N}d_{i}\,}
910:The standard deviation is estimated as
1010:
60:In adding these up, one finds that:
155:{\displaystyle d_{i}=x_{i}-x_{0}\,}
85:correct mean = 240 − 2 = 238.
13:
508:For instance with the exception,
307:{\displaystyle D={\frac {A}{N}}\,}
14:
1029:
817:The mean is then estimated to be
98:For a data set with assumed mean
73:7 + 4 cancels −6 − 5,
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24:is a method for calculating the
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522:Observed numbers in ranges
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10:
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500:Example using class ranges
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34:rapid calculation methods
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70:9 and −9 cancel,
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429:Bessel's correction
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1002:Mathematics portal
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30:standard deviation
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26:arithmetic mean
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22:assumed mean
21:
15:
530:tally-count
988:References
542:freq×diff
536:class diff
18:statistics
959:−
934:−
875:−
872:×
847:×
539:freq×diff
533:frequency
476:−
455:−
442:σ
391:−
378:σ
333:¯
234:∑
178:∑
139:−
105:suppose:
1012:Category
811:B = 371
778:186—188
759:183—185
733:180—182
704:177—179
670:174—176
641:171—173
615:168—170
590:165—167
567:162—164
547:159—161
808:A = −55
803:N = 100
79:average
40:Example
892:173.85
90:Method
20:, the
1018:Means
863:175.5
527:Range
317:Then
972:5.57
798:Sum
740:////
737:////
714:////
711:////
708:////
686:////
683:////
680:////
677:////
674:////
651:////
648:////
645:////
624:///
622:////
619:////
597:////
594:////
571:////
28:and
886:100
793:32
781://
754:44
728:16
665:16
662:−16
636:52
633:−26
610:90
607:−30
585:96
582:−24
562:25
16:In
1014::
878:55
773:0
751:22
745:11
742:/
725:16
719:16
716:/
699:0
690:25
659:−1
656:16
653:/
630:−2
627:13
604:−3
601:10
579:−4
573:/
559:−5
556:−5
550:/
431::
969:=
962:1
956:N
949:N
944:2
940:A
931:B
924:S
921:C
889:=
882:/
869:3
866:+
860:=
855:N
852:A
844:S
841:C
838:+
833:0
829:x
790:8
787:4
784:2
770:0
767:3
764:0
748:2
722:1
696:0
693:0
576:6
553:1
479:1
473:N
466:2
462:D
458:N
452:B
445:=
408:N
402:2
398:D
394:N
388:B
381:=
354:D
351:+
346:0
342:x
338:=
330:x
299:N
296:A
291:=
288:D
264:2
259:i
255:d
249:N
244:1
241:=
238:i
230:=
227:B
203:i
199:d
193:N
188:1
185:=
182:i
174:=
171:A
147:0
143:x
134:i
130:x
126:=
121:i
117:d
103:0
100:x
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