149:, as outliers change it significantly. Indeed, for many distributions it is one of the least efficient and least robust statistics. However, it finds some use in special cases: it is the maximally efficient estimator for the center of a uniform distribution, trimmed mid-ranges address robustness, and as an
340:
Conversely, for the normal distribution, the sample mean is the UMVU estimator of the mean. Thus for platykurtic distributions, which can often be thought of as between a uniform distribution and a normal distribution, the informativeness of the middle sample points versus the extrema values varies
331:
and sample minimum, together with sample size, are a sufficient statistic for the population maximum and minimum – the distribution of other samples, conditional on a given maximum and minimum, is just the uniform distribution between the maximum and minimum and thus add no information. See
336:
for further discussion. Thus the mid-range, which is an unbiased and sufficient estimator of the population mean, is in fact the UMVU: using the sample mean just adds noise based on the uninformative distribution of points within this range.
539:
609:
133:
defined as the difference between maximum and minimum values. The two measures are complementary in sense that if one knows the mid-range and the range, one can find the sample maximum and minimum values.
120:
709:
341:
from "equal" for normal to "uninformative" for uniform, and for different distributions, one or the other (or some combination thereof) may be most efficient. A robust analog is the
169:
of 0, meaning that a single observation can change it arbitrarily. Further, it is highly influenced by outliers: increasing the sample maximum or decreasing the sample minimum by
278:
can be interpreted as the fully trimmed (50%) mid-range; this accords with the convention that the median of an even number of points is the mean of the two middle points.
233:
202:
472:
324:
902:
879:
856:
622:
562:
377:. The following table summarizes empirical data comparing three estimators of the mean for distributions of varied kurtosis; the
435:= 1 or 2, the midrange and the mean are equal (and coincide with the median), and are most efficient for all distributions. For
320:
293:: differences of midsummaries, such as midhinge minus the median, give measures of skewness at different points in the tail.
439:= 3, the modified mean is the median, and instead the mean is the most efficient measure of central tendency for values of
459:
927:
658:
75:
668:
142:
161:
The midrange is highly sensitive to outliers and ignores all but two data points. It is therefore a very non-
634:
302:
650:
282:
130:
794:
638:
549:
712:
333:
58:
724:
126:
898:
875:
852:
662:
239:
35:
31:
162:
54:
358:
263:
207:
176:
166:
62:
922:
868:
845:
382:
328:
235:
It is thus of little use in practical statistics, unless outliers are already handled.
916:
891:
614:
and, in particular, the variance does not decrease to zero as the sample size grows.
378:
773:
778:(Master's). University of North Carolina at Chapel Hill. Table (4.1), pp. 32–34.
309:
286:
150:
204:
while it changes the sample mean, which also has breakdown point of 0, by only
313:
42:
305:
301:
Despite its drawbacks, in some cases it is useful: the midrange is a highly
137:
The mid-range is rarely used in practical statistical analysis, as it lacks
357:
from 4 to 20) drawn from a sufficiently platykurtic distribution (negative
17:
729:
290:
271:
66:
775:
An
Investigation of Measures of Central Tendency Used in Quality Control
373:)²) − 3), the mid-range is an efficient estimator of the mean
342:
654:
534:{\displaystyle \operatorname {var} (M)={\frac {\pi ^{2}}{24\ln(n)}}.}
275:
145:
of interest, because it ignores all intermediate points, and lacks
649:
While the mean of a set of values minimizes the sum of squares of
345:, which averages the midhinge (25% trimmed mid-range) and median.
893:
Applications, Basics and
Computing of Exploratory Data Analysis
604:{\displaystyle \operatorname {var} (M)={\frac {\pi ^{2}}{12}}}
323:
with unknown maximum and minimum, the mid-range is the
671:
565:
475:
210:
179:
78:
890:
867:
844:
703:
603:
533:
227:
196:
114:
672:
385:, where the maximum and minimum are eliminated.
281:These trimmed midranges are also of interest as
97:
88:
759:
27:Arithmetic mean of the maximum and the minimum
831:
819:
807:
446:from 2.0 to 6.0 as well as from −0.8 to 2.0.
325:uniformly minimum-variance unbiased estimator
308:of μ, given a small sample of a sufficiently
262:)% percentiles, and is more robust, having a
115:{\displaystyle M={\frac {\max x+\min x}{2}}.}
8:
870:The Advanced Theory of Statistics, Volume 1
847:The Oxford dictionary of Statistical Terms
556:is unbiased, and has a variance given by:
466:is unbiased, and has a variance given by:
153:, it is simple to understand and compute.
704:{\displaystyle \max \left|x_{i}-m\right|}
684:
670:
590:
584:
564:
500:
494:
474:
254:% trimmed midrange is the average of the
214:
209:
183:
178:
85:
77:
65:of the maximum and minimum values of the
889:Velleman, P. F.; Hoaglin, D. C. (1981).
387:
312:distribution, but it is inefficient for
125:The mid-range is closely related to the
740:
791:Statistical methods in quality control
747:
7:
327:(UMVU) estimator for the mean. The
316:distributions, such as the normal.
274:, which is the 25% midsummary. The
866:Kendall, M.G.; Stuart, A. (1969).
25:
789:Cowden, Dudley Johnstone (1957).
270:%. In the middle of these is the
772:Vinson, William Daniel (1951).
321:continuous uniform distribution
146:
138:
578:
572:
522:
516:
488:
482:
1:
661:, the midrange minimizes the
460:standard normal distribution
398:Most efficient estimator of
851:. Oxford University Press.
760:Velleman & Hoaglin 1981
944:
793:. Prentice-Hall. pp.
659:average absolute deviation
29:
832:Kendall & Stuart 1969
820:Kendall & Stuart 1969
808:Kendall & Stuart 1969
711:): it is a solution to a
173:changes the mid-range by
141:as an estimator for most
353:For small sample sizes (
635:asymptotic distribution
289:of central location or
242:midrange is known as a
705:
605:
535:
283:descriptive statistics
229:
198:
131:statistical dispersion
116:
30:For loudspeakers, see
706:
617:For a sample of size
606:
544:For a sample of size
536:
454:For a sample of size
230:
199:
117:
34:. For computers, see
669:
639:Laplace distribution
623:uniform distribution
621:from a zero-centred
563:
550:Laplace distribution
473:
228:{\displaystyle x/n.}
208:
197:{\displaystyle x/2,}
177:
76:
713:variational problem
450:Sampling properties
334:German tank problem
319:For example, for a
928:Summary statistics
843:Dodge, Y. (2003).
725:Range (statistics)
701:
601:
548:from the standard
531:
391:Excess kurtosis (γ
225:
194:
112:
897:. Duxbury Press.
663:maximum deviation
599:
526:
429:
428:
258:% and (100−
107:
36:Midrange computer
32:Mid-range speaker
16:(Redirected from
935:
908:
896:
885:
873:
862:
850:
835:
834:, Example 14.12.
829:
823:
817:
811:
805:
799:
798:
786:
780:
779:
769:
763:
757:
751:
745:
710:
708:
707:
702:
700:
696:
689:
688:
625:, the mid-range
610:
608:
607:
602:
600:
595:
594:
585:
552:, the mid-range
540:
538:
537:
532:
527:
525:
505:
504:
495:
462:, the mid-range
388:
248:
247:
234:
232:
231:
226:
218:
203:
201:
200:
195:
187:
163:robust statistic
121:
119:
118:
113:
108:
103:
86:
55:central tendency
53:is a measure of
21:
943:
942:
938:
937:
936:
934:
933:
932:
913:
912:
911:
905:
888:
882:
865:
859:
842:
838:
830:
826:
822:, Example 14.5.
818:
814:
810:, Example 14.4.
806:
802:
788:
787:
783:
771:
770:
766:
758:
754:
746:
742:
738:
721:
680:
679:
675:
667:
666:
647:
586:
561:
560:
506:
496:
471:
470:
452:
445:
394:
372:
368:
364:
359:excess kurtosis
351:
299:
264:breakdown point
245:
244:
206:
205:
175:
174:
167:breakdown point
159:
129:, a measure of
87:
74:
73:
63:arithmetic mean
61:defined as the
39:
28:
23:
22:
15:
12:
11:
5:
941:
939:
931:
930:
925:
915:
914:
910:
909:
903:
886:
880:
863:
857:
839:
837:
836:
824:
812:
800:
781:
764:
752:
739:
737:
734:
733:
732:
727:
720:
717:
699:
695:
692:
687:
683:
678:
674:
657:minimizes the
646:
643:
612:
611:
598:
593:
589:
583:
580:
577:
574:
571:
568:
542:
541:
530:
524:
521:
518:
515:
512:
509:
503:
499:
493:
490:
487:
484:
481:
478:
451:
448:
443:
427:
426:
425:Modified mean
423:
419:
418:
415:
411:
410:
407:
403:
402:
396:
392:
383:truncated mean
370:
366:
362:
361:, defined as γ
350:
347:
329:sample maximum
298:
295:
224:
221:
217:
213:
193:
190:
186:
182:
158:
155:
123:
122:
111:
106:
102:
99:
96:
93:
90:
84:
81:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
940:
929:
926:
924:
921:
920:
918:
906:
904:0-87150-409-X
900:
895:
894:
887:
883:
881:0-85264-141-9
877:
872:
871:
864:
860:
858:0-19-920613-9
854:
849:
848:
841:
840:
833:
828:
825:
821:
816:
813:
809:
804:
801:
796:
792:
785:
782:
777:
776:
768:
765:
761:
756:
753:
749:
744:
741:
735:
731:
728:
726:
723:
722:
718:
716:
714:
697:
693:
690:
685:
681:
676:
664:
660:
656:
652:
644:
642:
640:
636:
632:
629:is unbiased,
628:
624:
620:
615:
596:
591:
587:
581:
575:
569:
566:
559:
558:
557:
555:
551:
547:
528:
519:
513:
510:
507:
501:
497:
491:
485:
479:
476:
469:
468:
467:
465:
461:
457:
449:
447:
442:
438:
434:
424:
421:
420:
416:
413:
412:
408:
405:
404:
401:
397:
390:
389:
386:
384:
380:
379:modified mean
376:
360:
356:
349:Small samples
348:
346:
344:
338:
335:
330:
326:
322:
317:
315:
311:
307:
304:
296:
294:
292:
288:
284:
279:
277:
273:
269:
265:
261:
257:
253:
249:
241:
236:
222:
219:
215:
211:
191:
188:
184:
180:
172:
168:
164:
156:
154:
152:
148:
144:
143:distributions
140:
135:
132:
128:
109:
104:
100:
94:
91:
82:
79:
72:
71:
70:
68:
64:
60:
56:
52:
48:
44:
37:
33:
19:
892:
869:
846:
827:
815:
803:
790:
784:
774:
767:
755:
743:
665:(defined as
648:
630:
626:
618:
616:
613:
553:
545:
543:
463:
455:
453:
440:
436:
432:
430:
414:−0.8 to 2.0
406:−1.2 to −0.8
399:
374:
354:
352:
339:
318:
300:
287:L-estimators
280:
267:
259:
255:
251:
243:
237:
170:
160:
136:
124:
50:
46:
40:
874:. Griffin.
637:which is a
422:2.0 to 6.0
310:platykurtic
165:, having a
151:L-estimator
51:mid-extreme
917:Categories
748:Dodge 2003
736:References
651:deviations
314:mesokurtic
297:Efficiency
246:midsummary
157:Robustness
147:robustness
139:efficiency
43:statistics
18:Half-range
691:−
645:Deviation
588:π
570:
514:
498:π
480:
458:from the
409:Midrange
306:estimator
303:efficient
47:mid-range
730:Midhinge
719:See also
653:and the
291:skewness
272:midhinge
67:data set
633:has an
381:is the
343:trimean
240:trimmed
901:
878:
855:
655:median
285:or as
276:median
250:– the
59:sample
45:, the
923:Means
795:67–68
417:Mean
127:range
57:of a
899:ISBN
876:ISBN
853:ISBN
431:For
365:= (μ
673:max
567:var
477:var
369:/(μ
266:of
98:min
89:max
49:or
41:In
919::
715:.
641:.
631:nM
597:12
511:ln
508:24
238:A
69::
907:.
884:.
861:.
797:.
762:.
750:.
698:|
694:m
686:i
682:x
677:|
627:M
619:n
592:2
582:=
579:)
576:M
573:(
554:M
546:n
529:.
523:)
520:n
517:(
502:2
492:=
489:)
486:M
483:(
464:M
456:n
444:2
441:γ
437:n
433:n
400:μ
395:)
393:2
375:μ
371:2
367:4
363:2
355:n
268:n
260:n
256:n
252:n
223:.
220:n
216:/
212:x
192:,
189:2
185:/
181:x
171:x
110:.
105:2
101:x
95:+
92:x
83:=
80:M
38:.
20:)
Text is available under the Creative Commons Attribution-ShareAlike License. Additional terms may apply.