Материал: chap2
ȼ ɞɚɧɧɨɣ ɝɪɭɩɩɟ ɢɡ 100 ɱɟɥɨɜɟɤ ɜɫɟ, ɭ ɤɨɝɨ ɜɵɫɨɤɚɹ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɬɪɭɞɚ, ɭɞɨɜɥɟɬɜɨɪɟɧɵ ɩɪɨɮɟɫɫɢɟɣ (ɧɨ ɧɟ ɧɚɨɛɨɪɨɬ! ɨɛ ɷɬɨɦ, ɜɩɪɨɱɟɦ, ɩɨɡɞɧɟɟ), ɬ.ɟ. ɢɦɟɟɬɫɹ ɩɨɥɧɚɹ ɨɩɪɟɞɟɥɟɧɧɨɫɬɶ ɨɬɧɨɫɢɬɟɥɶɧɨ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ ɩɪɨɮɟɫɫɢɟɣ ɭ ɜɫɟɯ ɪɚɛɨɬɧɢɤɨɜ ɫ ɜɵɫɨɤɨɣ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶɸ ɬɪɭɞɚ. Ʌɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɩɪɢ ɷɬɨɦ Q=1.
Ⱦɚɥɟɟ ɛɭɞɟɦ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɞɪɭɝɢɟ ɝɪɭɩɩɵ, ɞɥɹ ɤɨɬɨɪɵɯ ɤɨɪɪɟɥɹɰɢɨɧɧɵɟ ɬɚɛɥɢɰɵ ɢɦɟɸɬ ɬɟ ɠɟ ɦɚɪɝɢɧɚɥɵ, ɩɨɷɬɨɦɭ ɜɨɫɩɪɨɢɡɜɨɞɢɬɶ ɛɭɞɟɦ ɥɢɲɶ ɜɧɭɬɪɢɤɥɟɬɨɱɧɵɟ ɱɚɫɬɨɬɵ.
19 |
1 |
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15 |
5 |
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10 |
10 |
31 |
49 |
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35 |
45 |
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40 |
40 |
ɚ |
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ɛ |
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ɜ |
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ɇɚɩɪɢɦɟɪ, ɞɥɹ ɬɚɛɥɢɰɵ ɚ ɫɜɹɡɶ, ɨɱɟɜɢɞɧɨ, ɦɟɧɶɲɟ, ɦɟɧɶɲɢɦ ɨɤɚɡɵɜɚɟɬɫɹ ɢ Q=0,94. Ⱦɥɹ ɬɚɛɥɢɰɵ ɛ ɫɜɹɡɶ ɟɳɟ ɦɟɧɶɲɟ, ɢ Q=+0,59. Ⱥ ɞɥɹ ɬɚɛɥɢɰɵ ɜ ɫɜɹɡɢ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ ɧɟɬ: ɢ ɭ ɪɚɛɨɬɧɢɤɨɜ ɫ ɜɵɫɨɤɨɣ, ɢ ɭ ɪɚɛɨɬɧɢɤɨɜ ɫ ɧɢɡɤɨɣ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶɸ ɬɪɭɞɚ ɱɢɫɥɚ ɭɞɨɜɥɟɬɜɨɪɟɧɧɵɯ ɢ ɧɟɭɞɨɜɥɟɬɜɨɪɟɧɧɵɯ ɩɪɨɮɟɫɫɢɟɣ ɨɞɢɧɚɤɨɜɵ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ ɨɛɪɚɳɚɟɬɫɹ ɜ ɧɭɥɶ ɢ Q.
Ⱦɥɹ ɬɨɝɨ ɱɬɨɛɵ |Q| ɛɵɥ ɪɚɜɟɧ 1, ɞɨɫɬɚɬɨɱɧɨ, ɱɬɨɛɵ ɨɞɧɚ ɢɡ ɜɧɭɬɪɢɤɥɟɬɨɱɧɵɯ ɱɚɫɬɨɬ ɨɛɪɚɬɢɥɚɫɶ ɜ ɧɭɥɶ. ɇɚɩɪɢɦɟɪ, ɩɪɢ N12=0 |Q|=1. ɗɬɨ ɡɧɚɱɢɬ, ɱɬɨ ɟɫɥɢ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɜɵɫɨɤɚɹ, ɬɨ ɨɛɹɡɚɬɟɥɶɧɨ ɭɞɨɜɥɟɬɜɨɪɟɧ (ɪɚɡɭɦɟɟɬɫɹ, ɪɟɱɶ ɢɞɟɬ ɫɞɚɧɧɨɣ ɝɢɩɨɬɟɬɢɱɟɫɤɨɣ ɝɪɭɩɩɟ) ɩɪɨɮɟɫɫɢɟɣ. Ɉɛɪɚɬɧɨɟ ɧɟɜɟɪɧɨ: ɟɫɥɢ ɭɞɨɜɥɟɬɜɨɪɟɧ, ɬɨ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɢ ɜɵɫɨɤɚɹ ɢ ɧɢɡɤɚɹ. ɋɥɟɞɨɜɚɬɟɥɶɧɨ, Q — ɩɨɤɚɡɚɬɟɥɶ ɨɞɧɨɫɬɨɪɨɧɧɟɣ ɫɜɹɡɢ. ȿɫɥɢ ɦɟɠɞɭ ɡɧɚɱɟɧɢɹɦɢ ɩɪɢɡɧɚɤɨɜ
Ɍɚɛɥɢɰɚ 25
Ɂɚɜɢɫɢɦɨɫɬɶ ɦɟɠɞɭ ɭɱɟɛɨɣ ɢ ɭɱɚɫɬɢɟɦ ɜ ɪɚɰɢɨɧɚɥɢɡɚɰɢɢ
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ɍɱɚɫɬɢɟ ɜ ɪɚɰɢɨɧɚɥɢɡɚɰɢɢ |
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Ɂɚɧɹɬɢɟ ɭɱɟɛɨɣ |
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ȼɫɟɝɨ |
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ɍɱɚɫɬɜɭɸɬ |
ɇɟ ɭɱɚɫɬɜɭɸɬ |
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ɭɱɚɬɫɹ |
29 |
93 |
122 |
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ɧɟ ɭɱɚɬɫɹ |
5 |
93 |
98 |
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34 |
186 |
120 |
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[87]
ɞɨɩɭɫɬɢɦɨ ɭɩɨɪɹɞɨɱɟɧɢɟ, ɤɚɤ ɜ ɧɚɲɟɦ ɩɪɢɦɟɪɟ, ɬɨ Q>0 ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɩɪɹɦɨɣ (ɜɵɫɨɤɨɣ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɢ ɨɬɜɟɱɚɟɬ ɜɵɫɨɤɚɹ ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɶ), a Q<0 — ɨɛɪɚɬɧɨɣ ɫɜɹɡɢ.
ɍɩɪɚɠɧɟɧɢɟ 36. ȼɵɱɢɫɥɢɬɶ Q ɞɥɹ ɬɚɛɥɢɰɵ 25 (Ɉɬɜɟɬ: Q=0,71). ɋɜɹɡɶ ɟɫɬɶ. Ɉɧɚ ɨɞɧɨɫɬɨɪɨɧɧɹɹ: ɭɱɟɛɚ ɜɥɢɹɟɬ ɧɚ ɭɱɚɫɬɢɟ ɜ ɪɚɰɢɨɧɚɥɢɡɚɰɢɢ. ɗɬɨ ɠɟ ɩɨɞɬɜɟɪɠɞɚɟɬ ɡɧɚɱɟɧɢɟ ĭ (ɫɦ. ɧɢɠɟ).
Ʉɨɷɮɮɢɰɢɟɧɬ ɤɨɧɬɢɧɝɟɧɰɢɢ Ɏ ɩɨ ɨɩɪɟɞɟɥɟɧɢɸ:
N 11 N 22 N 12 N 21
Ɏ
N ( x 1 ) N ( x 2 ) N ( y1 ) N ( y 2 )
ȼ ɨɬɥɢɱɢɟ ɨɬ Q, ɤɨɬɨɪɵɣ ɨɛɪɚɳɚɟɬɫɹ ɜ ±1, ɤɨɝɞɚ ɯɨɬɹ ɛɵ ɨɞɧɚ ɜɧɭɬɪɢɤɥɟɬɨɱɧɚɹ ɱɚɫɬɨɬɚ ɪɚɜɧɚ ɧɭɥɸ, ɨɛɪɚɳɚɟɬɫɹ ɜ +1, ɤɨɝɞɚ N12=N2l=0, ɬ.ɟ. ɟɫɥɢ — ɜ ɧɚɲɟɦ ɩɪɢɦɟɪɟ — ɜɫɟ ɭɞɨɜɥɟɬɜɨɪɟɧɧɵɟ ɩɪɨɮɟɫɫɢɟɣ ɢɦɟɸɬ ɜɵɫɨɤɭɸ ɩɪɨɢɡɜɨɞɢɬɟɥɶɧɨɫɬɶ, ɚ ɧɟɭɞɨɜɥɟɬɜɨɪɟɧɧɵɟ — ɧɢɡɤɭɸ (ɢ ɧɚɨɛɨɪɨɬ!). Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ĭ ɹɜɥɹɟɬɫɹ ɩɨɤɚɡɚɬɟɥɟɦ ɞɜɭɫɬɨɪɨɧɧɟɣ ɫɜɹɡɢ. ɋɨɨɬɜɟɬɫɬɜɟɧɧɨ: ) d Q . ȿɫɥɢ ) t 0,5 , ɬɨ ɫɱɢɬɚɸɬ, ɱɬɨ ɧɚɞɟɠɧɨ ɭɫɬɚɧɨɜɥɟɧɚ ɞɜɭɫɬɨɪɨɧɧɹɹ
ɫɜɹɡɶ14. ȿɫɥɢ ɧɢɡɤɨɟ ɡɧɚɱɟɧɢɟ |
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ɨɬɜɟɱɚɟɬ ɨɬɫɭɬɫɬɜɢɸ ɫɜɹɡɢ ( |
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Q |
max |
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1) , ɬɨ ɧɢɡɤɨɟ ɡɧɚɱɟɧɢɟ |
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) ɦɨɠɟɬ ɛɵɬɶ ɫɥɟɞɫɬɜɢɟɦ ɦɚɪɝɢɧɚɥɶɧɨɝɨ ɷɮɮɟɤɬɚ: )max ɱɚɫɬɨ ɦɟɧɶɲɟ 1 (ɜ ɷɬɨɦ ɦɨɠɧɨ
14 Ȼɨɥɟɟ ɫɬɪɨɝɨ ɡɧɚɱɢɦɨɫɬɶ ĭ ɢ Q ɨɩɪɟɞɟɥɹɸɬ ɫ ɩɨɦɨɳɶɸ ɤɪɢɬɟɪɢɹ Ȥ2.
59
ɭɛɟɞɢɬɶɫɹ ɧɚ ɩɪɢɦɟɪɚɯ). ɍ ɪɚɡɧɵɯ ɬɚɛɥɢɰ ɪɚɡɧɵɟ ĭmax, ɩɨɷɬɨɦɭ ĭ, ɪɚɫɫɱɢɬɚɧɧɵɟ ɞɥɹ ɧɢɯ, ɱɚɫɬɨ ɧɟɫɨɩɨɫɬɚɜɢɦɵ.
Ɇɨɠɧɨ ɩɨɤɚɡɚɬɶ, ɱɬɨ |
ɧɨɪɦɢɪɨɜɤɚ ĭ (ɩɟɪɟɯɨɞ |
ɤ )' |
) |
) ɛɵɥɚ ɛɵ ɧɟɡɚɤɨɧɧɵɦ |
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)max |
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ɭɫɢɥɟɧɢɟɦ ɩɨɤɚɡɚɬɟɥɹ ɫɜɹɡɢ. ȿɫɥɢ ĭ ɦɚɥ, ɜɵɱɢɫɥɹɸɬ Q, ɱɬɨɛɵ ɭɫɬɚɧɨɜɢɬɶ, ɟɫɬɶ ɥɢ ɯɨɬɹ ɛɵ |
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ɨɞɧɨɫɬɨɪɨɧɧɹɹ ɫɜɹɡɶ. Ɍɚɤ, |
ɞɥɹ ɬɚɛɥɢɰɵ 25 ĭ =0,26, |
ɚ Q=0,71. Ɇɨɠɧɨ ɫɱɢɬɚɬɶ ɧɚɞɟɠɧɨ |
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ɭɫɬɚɧɨɜɥɟɧɧɨɣ ɨɞɧɨɫɬɨɪɨɧɧɸɸ ɫɜɹɡɶ. (ȼɵɱɢɫɥɟɧɢɟ ɷɬɢɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɫɨɫɬɚɜɥɹɟɬ ɫɨɞɟɪɠɚɧɢɟ ɭɩɪɚɠɧɟɧɢɹ 37).
ɉɪɢɜɟɞɟɦ ɩɪɢɦɟɪɵ ɩɪɢɦɟɧɟɧɢɹ Q ɢ ĭ ɜ ɫɨɰɢɚɥɶɧɵɯ ɢɫɫɥɟɞɨɜɚɧɢɹɯ (ɬɚɤ ɤɚɤ ɜɵɱɢɫɥɟɧɢɹ ɤɨɷɮɮɢɰɢɟɧɬɨɜ ɩɪɢɜɨɞɢɬɶɫɹ ɧɟ ɛɭɞɭɬ, ɤɚɠɞɵɣ ɢɡ ɪɚɡɛɢɪɚɟɦɵɯ ɩɪɢɦɟɪɨɜ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɤɚɤ ɱɚɫɬɶ ɭɩɪɚɠɧɟɧɢɹ 38). ɉɭɫɬɶ X — ɦɟɫɬɨ ɩɪɨɠɢɜɚɧɢɹ, x1 — ɝɨɪɨɞ, ɯ2 — ɫɟɥɶɫɤɚɹ ɦɟɫɬɧɨɫɬɶ, a Y — ɭɪɨɜɟɧɶ ɨɛɪɚɡɨɜɚɧɢɹ, ɭ1 — ɜɵɫɲɟɟ, ɫɪɟɞɧɟɟ (ɨɤɨɧɱɟɧɧɨɟ ɢ ɧɟɨɤɨɧɱɟɧɧɨɟ), ɭ2 — ɧɚɱɚɥɶɧɨɟ (ɨɤɨɧɱɟɧɧɨɟ ɢ ɧɟɨɤɨɧɱɟɧɧɨɟ). ȼ ɬɚɛɥɢɰɚɯ, ɤɨɬɨɪɵɟ ɦɵ ɩɪɢɜɟɞɟɦ ɩɨ ɤɧɢɝɟ Ɏ.Ɇ. Ȼɨɪɨɞɤɢɧɚ «ɋɬɚɬɢɫɬɢɱɟɫɤɚɹ ɨɰɟɧɤɚ ɫɜɹɡɟɣ ɦɟɠɞɭ ɷɤɨɧɨɦɢɱɟ-
[88]
ɫɤɢɦɢ ɩɨɤɚɡɚɬɟɥɹɦɢ» (Ɇ., 1968), ɤɨɥɢɱɟɫɬɜɚ ɜɵɪɚɠɟɧɵ ɜ ɦɢɥɥɢɨɧɚɯ ɱɟɥɨɜɟɤ. ɂɬɚɤ, ɩɨ ɞɚɧɧɵɦ ɧɚ 1939 ɝ.:
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Y |
N(xi) |
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y1 |
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y2 |
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x1 |
10,76 |
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45,34 |
56,10 |
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x2 |
5,10 |
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109,40 |
114,50 |
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N(yj) |
15,86 |
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154,74 |
170,60 |
Ɋɚɫɩɪɟɞɟɥɟɧɢɟ ɦɚɪɝɢɧɚɥɨɜ ɫɯɨɞɧɨ, ) =0,24; Q=0,67 ɋɜɹɡɶ ɟɫɬɶ, ɨɧɚ ɫɭɳɟɫɬɜɟɧɧɚɹ, ɨɞɧɨɫɬɨɪɨɧɧɹɹ (ɟɫɥɢ ɠɢɬɟɥɶ ɫɟɥɶɫɤɨɣ ɦɟɫɬɧɨɫɬɢ, ɬɨ ɜ ɛɨɥɶɲɢɧɫɬɜɟ ɫɥɭɱɚɟɜ — ɱɟɥɨɜɟɤ ɫ ɧɢɡɤɢɦ ɨɛɪɚɡɨɜɚɬɟɥɶɧɵɦ ɭɪɨɜɧɟɦ).
ɉɨ ɞɚɧɧɵɦ ɡɚ 1959 ɝ.:
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Y |
N(xi) |
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y1 |
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y2 |
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x1 |
37,63 |
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62,17 |
99,80 |
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x2 |
21,08 |
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87,92 |
109,00 |
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N(yj) |
58,71 |
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150,09 |
208,80 |
Ɍɟɩɟɪɶ Q=0,43. ɗɬɨ ɦɟɧɶɲɟ, ɱɟɦ Q ɞɥɹ ɩɪɟɞɵɞɭɳɟɣ ɬɚɛɥɢɰɵ (1939). ɋɥɟɞɨɜɚɬɟɥɶɧɨ, ɤɚɤ ɜɢɞɢɦ, ɪɚɡɥɢɱɢɹ ɜ ɭɪɨɜɧɟ ɨɛɪɚɡɨɜɚɧɢɹ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɫɬɢɪɚɸɬɫɹ, ɯɨɬɹ ɢ ɨɫɬɚɸɬɫɹ.
ɉɨ ɞɚɧɧɵɦ 1939 ɝ. ɢ 1959 ɝ. ɩɪɨɫɥɟɞɢɦ ɫɜɹɡɶ ɦɟɠɞɭ ɨɛɫɭɠɞɚɟɦɵɦɢ ɩɪɢɡɧɚɤɚɦɢ ɭ ɦɭɠɱɢɧ ɢ ɭ ɠɟɧɳɢɧ ɜ ɨɬɞɟɥɶɧɨɫɬɢ.
Ⱦɥɹ ɦɭɠɱɢɧ ɫɨɨɬɜɟɬɫɬɜɭɸɳɚɹ ɬɚɛɥɢɰɚ (1939 ɝ.):
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X |
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Y |
N(xi) |
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y1 |
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y2 |
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x1 |
5,58 |
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23,32 |
28,90 |
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x2 |
3,27 |
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59,23 |
62,50 |
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N(yj) |
8,85 |
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82,55 |
91,40 |
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Q=0,63 |
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[89] |
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Ⱦɥɹ ɠɟɧɳɢɧ: |
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X |
Y |
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N(xi) |
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y1 |
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y2 |
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x1 |
5,18 |
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26,31 |
31,49 |
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x2 |
1,83 |
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65,95 |
67,78 |
60
N(yj) |
7,01 |
92,26 |
99,27 |
Q=0,75
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɪɚɡɥɢɱɢɟ ɜ ɭɪɨɜɧɟ ɨɛɪɚɡɨɜɚɧɢɹ ɝɨɪɨɠɚɧɨɤ ɢ ɫɟɥɶɫɤɢɯ ɠɢɬɟɥɶɧɢɰ ɛɨɥɟɟ ɫɭɳɟɫɬɜɟɧɧɨɟ, ɱɟɦ ɭ ɦɭɠɱɢɧ.
ɉɪɨɫɥɟɞɢɦ ɞɢɧɚɦɢɤɭ. ɂɡ ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɬɚɛɥɢɰ (ɞɚɧɧɵɟ 1959 ɝ.) ɞɥɹ ɦɭɠɱɢɧ Q=0,38, ɞɥɹ ɠɟɧɳɢɧ Q=0,47. ɋɞɟɥɚɧɧɵɣ ɪɚɧɟɟ ɜɵɜɨɞ ɫɨɯɪɚɧɹɟɬɫɹ, ɧɨ ɫɜɹɡɶ ɫɬɚɧɨɜɢɬɫɹ ɦɟɧɟɟ ɫɭɳɟɫɬɜɟɧɧɨɣ: ɢ ɭ ɦɭɠɱɢɧ, ɢ ɭ ɠɟɧɳɢɧ ɫ ɬɟɱɟɧɢɟɦ ɜɪɟɦɟɧɢ ɫɬɢɪɚɸɬɫɹ ɪɚɡɥɢɱɢɹ ɨɛɪɚɡɨɜɚɬɟɥɶɧɨɝɨ ɭɪɨɜɧɹ ɝɨɪɨɠɚɧ ɢ ɫɟɥɶɫɤɢɯ ɠɢɬɟɥɟɣ, ɯɨɬɹ ɭ ɠɟɧɳɢɧ ɷɬɢ ɪɚɡɥɢɱɢɹ ɨɫɬɚɸɬɫɹ ɧɟɫɤɨɥɶɤɨ ɛɨɥɶɲɢɦɢ.
Ⱥ ɬɟɩɟɪɶ ɨɛɪɚɬɢɦɫɹ ɤ ɦɚɬɟɪɢɚɥɚɦ ɩɟɪɟɩɢɫɢ 1970 ɝ. ȼ III ɬɨɦɟ «ɂɬɨɝɨɜ ɜɫɟɫɨɸɡɧɨɣ ɩɟɪɟɩɢɫɢ ɧɚɫɟɥɟɧɢɹ 1970 ɝɨɞɚ» — «ɍɪɨɜɟɧɶ ɨɛɪɚɡɨɜɚɧɢɹ ɧɚɫɟɥɟɧɢɹ ɋɋɋɊ» (Ɇɨɫɤɜɚ, 1972, ɫ. 206) — ɩɪɢɜɨɞɹɬɫɹ ɬɚɤɢɟ ɞɚɧɧɵɟ: ɧɚ 1000 ɱɟɥɨɜɟɤ ɝɨɪɨɞɫɤɨɝɨ ɧɚɫɟɥɟɧɢɹ ɩɪɢɯɨɞɢɬɫɹ 592 ɱɟɥ. ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɜɵɲɟ ɧɚɱɚɥɶɧɨɝɨ, ɧɚ 1000 ɠɟ ɱɟɥɨɜɟɤ ɫɟɥɶɫɤɨɝɨ ɧɚɫɟɥɟɧɢɹ — 332. Ɉɱɟɜɢɞɧɨ, ɩɨ ɷɬɢɦ ɞɚɧɧɵɦ ɧɟɥɶɡɹ ɧɟɩɨɫɪɟɞɫɬɜɟɧɧɨ ɪɚɫɫɱɢɬɚɬɶ Q, ɬɚɤ ɤɚɤ ɱɢɫɥɟɧɧɨɫɬɶ ɝɨɪɨɞɫɤɨɝɨ ɢ ɫɟɥɶɫɤɨɝɨ ɧɚɫɟɥɟɧɢɹ ɧɟɨɞɢɧɚɤɨɜɚ.
ɉɨ ɞɚɧɧɵɦ V ɬɨɦɚ «ɉɟɪɟɩɢɫɢ» ɜ ɝɨɪɨɞɚɯ ɩɪɨɠɢɜɚɥɨ 135,33, ɚ ɜ ɫɟɥɚɯ—106,11 ɦɢɥɥɢɨɧɨɜ ɱɟɥɨɜɟɤ. ɇɭɠɧɨ, ɨɱɟɜɢɞɧɨ, 135,33 ɪɚɡɞɟɥɢɬɶ ɜ ɨɬɧɨɲɟɧɢɢ 592:408, ɚ 106,11 — ɜ ɨɬɧɨɲɟɧɢɢ 332:668. ȼ ɪɟɡɭɥɶɬɚɬɟ ɩɨɥɭɱɚɟɦ ɬɚɛɥɢɰɭ:
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X |
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Y |
N(xi) |
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y1 |
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y2 |
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x1 |
80,12 |
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55,21 |
135,33 |
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x2 |
35,23 |
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70,88 |
106,11 |
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N(yj) |
115,35 |
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126,09 |
241,44 |
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Q=0,49 |
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[90]
ɍɩɪɚɠɧɟɧɢɟ 39. Ɇɭɠɫɤɨɟ ɧɚɫɟɥɟɧɢɟ ɝɨɪɨɞɨɜ ɫɨɫɬɚɜɥɹɟɬ 62,68 ɦɥɧ. ɱɟɥ., ɫɟɥɶɫɤɨɣ ɦɟɫɬɧɨɫɬɢ — 48,50. ɇɚ 1000 ɦɭɠɱɢɧ, ɩɪɨɠɢɜɚɸɳɢɯ ɜ ɝɨɪɨɞɟ, ɩɪɢɯɨɞɢɬɫɹ 621 ɱɟɥ. ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɜɵɲɟ ɧɚɱɚɥɶɧɨɝɨ, ɚ ɜ ɫɟɥɶɫɤɨɣ ɦɟɫɬɧɨɫɬɢ — 388 ɱɟɥ.
ɋɨɫɬɚɜɢɬɶ ɬɚɛɥɢɰɭ, ɜɵɱɢɫɥɢɬɶ Q.
Ɉɬɜɟɬ: Q=0,44.
ɍɩɪɚɠɧɟɧɢɟ 40. ɀɟɧɫɤɨɟ ɧɚɫɟɥɟɧɢɟ ɝɨɪɨɞɨɜ ɫɨɫɬɚɜɥɹɟɬ 72,65 ɦɥɧ. ɱɟɥ., ɫɟɥɶɫɤɨɟ — 57,60 ɦɥɧ. ɱɟɥ. ɇɚ 1000ɠɟɧɳɢɧ, ɩɪɨɠɢɜɚɸɳɢɯ ɜ ɝɨɪɨɞɚɯ, ɩɪɢɯɨɞɢɬɫɹ 568 ɱɟɥ. ɫ ɨɛɪɚɡɨɜɚɧɢɟɦ ɜɵɲɟ ɧɚɱɚɥɶɧɨɝɨ, ɜ ɫɟɥɶɫɤɨɣ — 296. ɋɨɫɬɚɜɢɬɶ ɬɚɛɥɢɰɭ, ɜɵɱɢɫɥɢɬɶ Q.
Ɉɬɜɟɬ: Q=0,52.
Ⱦɥɹ ɤɨɧɬɪɨɥɹ ɜɫɟɯ ɬɚɛɥɢɰ: ɜɫɟ ɧɚɫɟɥɟɧɢɟ ɋɋɋɊ ɜ 1970 ɝ. ɫɨɫɬɚɜɥɹɥɨ 241,44 ɦɥɧ. ɱɟɥ., ɜ ɬɨɦ ɱɢɫɥɟ: ɠɟɧɳɢɧ — 130,26 ɦɥɧ. ɱɟɥ., ɦɭɠɱɢɧ — 111,18 ɦɥɧ. ɱɟɥ.
Ɋɚɫɫɦɨɬɪɢɦ ɩɨɥɭɱɟɧɧɵɟ ɪɟɡɭɥɶɬɚɬɵ. Ƚɪɚɦɨɬɧɨɫɬɶ ɧɚɫɟɥɟɧɢɹ ɋɋɋɊ ɧɟɭɤɥɨɧɧɨ ɜɨɡɪɚɫɬɚɟɬ, ɨɞɧɚɤɨ ɪɚɡɥɢɱɢɟ ɜ ɭɪɨɜɧɟ ɨɛɪɚɡɨɜɚɧɢɹ ɠɢɬɟɥɟɣ ɝɨɪɨɞɨɜ ɢ ɫɟɥɶɫɤɢɯ ɦɟɫɬɧɨɫɬɟɣ ɨɫɬɚɸɬɫɹ: ɬɟɦɩɵ ɪɨɫɬɚ ɨɛɪɚɡɨɜɚɬɟɥɶɧɨɝɨ ɭɪɨɜɧɹ ɜ ɝɨɪɨɞɚɯ ɜɵɲɟ.
ɇɟɤɨɬɨɪɨɟ ɭɜɟɥɢɱɟɧɢɟ Q ɞɥɹ ɬɚɛɥɢɰ 1970 ɝ. ɩɨ ɫɪɚɜɧɟɧɢɸ ɫ Q ɞɥɹ ɬɚɛɥɢɰ 1959 ɝ. ɫɜɹɡɚɧɨ, ɩɨ-ɜɢɞɢɦɨɦɭ, ɫ ɩɪɨɞɨɥɠɚɸɳɢɦɫɹ ɨɬɬɨɤɨɦ ɦɨɥɨɞɟɠɢ ɢɡ ɫɟɥɶɫɤɢɯ ɦɟɫɬɧɨɫɬɟɣ ɜ ɝɨɪɨɞɚ. ɂɡ ɫɟɥɚ ɭɯɨɞɹɬ ɩɪɟɢɦɭɳɟɫɬɜɟɧɧɨ ɦɨɥɨɞɵɟ ɥɸɞɢ ɫɨ ɫɪɟɞɧɢɦ (ɨɤɨɧɱɟɧɧɵɦ ɢ ɧɟɨɤɨɧɱɟɧɧɵɦ) ɨɛɪɚɡɨɜɚɧɢɟɦ, ɜ ɫɟɥɟ, ɬɚɤɢɦ ɨɛɪɚɡɨɦ, ɭɜɟɥɢɱɢɜɚɟɬɫɹ ɞɨɥɹ ɬɟɯ, ɭ ɤɨɝɨ ɨɛɪɚɡɨɜɚɧɢɟ ɧɟ ɜɵɲɟ ɧɚɱɚɥɶɧɨɝɨ (ɷɬɨ, ɜ ɨɫɧɨɜɧɨɦ, ɫɬɚɪɲɢɟ ɜɨɡɪɚɫɬɧɵɟ ɝɪɭɩɩɵ ɧɚɫɟɥɟɧɢɹ)15.
ɋɞɟɥɚɟɦ ɨɞɧɨ ɨɱɟɧɶ ɫɭɳɟɫɬɜɟɧɧɨɟ ɡɚɦɟɱɚɧɢɟ. ɂɡɭɱɚɟɦɵɟ ɫɨɰɢɨɥɨɝɚɦɢ ɫɨɜɨɤɭɩɧɨɫɬɢ ɱɚɫɬɨ ɨɤɚɡɵɜɚɸɬɫɹ ɜɟɫɶɦɚ ɪɚɡɧɨɪɨɞɧɵɦɢ. ɇɚɩɪɢɦɟɪ, ɪɚɛɨɱɢɟ ɩɪɟɞɩɪɢɹɬɢɹ —ɥɸɞɢ ɪɚɡɧɵɯ ɩɪɨɮɟɫɫɢɣ, ɪɚɡɧɨɝɨ ɩɨɥɚ, ɜɨɡɪɚɫɬɚ, ɨɛɪɚɡɨɜɚɧɢɹ ɢ ɬ.ɞ. ɉɪɢ ɞɨɫɬɚɬɨɱɧɨ ɪɚɡɧɨɪɨɞɧɨɣ
15 Ʌɸɛɨɩɵɬɧɵɣ ɩɪɢɦɟɪ ɩɪɢɦɟɧɟɧɢɹ Q ɜ ɫɨɰɢɨɥɨɝɢɢ ɱɢɬɚɬɟɥɶ ɦɨɠɟɬ ɧɚɣɬɢ ɜ ɫɬɚɬɶɟ ɋ. ɀɟɥɟɡɤɨ «Ɏɚɤɬɨɪɵ ɫɬɚɛɢɥɢɡɚɰɢɢ ɤɚɞɪɨɜ ɧɚ ɫɬɪɨɢɬɟɥɶɫɬɜɟ ȻȺɆɚ» (ɋɨɰɢɨɥɨɝɢɱɟɫɤɢɟ ɢɫɫɥɟɞɨɜɚɧɢɹ, 1980, ʋ 1, ɫ. 84—87).
61
ɫɨɜɨɤɭɩɧɨɫɬɢ ɦɨɝɭɬ ɜɨɡɧɢɤɚɬɶ ɤɚɠɭɳɢɟɫɹ ɫɜɹɡɢ, ɥɢɛɨ ɨɤɚɡɚɬɶɫɹ ɫɤɪɵɬɵɦɢ ɞɟɣɫɬɜɢɬɟɥɶɧɵɟ. ɉɨɹɫɧɢɦ ɷɬɨ ɩɪɢɦɟɪɨɦ.
ɉɪɢɦɟɪ 17. Ⱦɨɩɭɫɬɢɦ, ɱɬɨ ɧɟɤɨɬɨɪɚɹ ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɨɩɢɫɚɧɚ ɫ ɩɨɦɨɳɶɸ ɤɨɪɪɟɥɹɰɢɨɧɧɨɣ ɬɚɛɥɢɰɵ ɬɚɤɨɝɨ ɜɢɞɚ:
[91]
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X |
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Y |
N(xi) |
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y1 |
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y2 |
||
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x1 |
300 |
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300 |
600 |
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x2 |
200 |
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200 |
400 |
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N(yj) |
500 |
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500 |
1000 |
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Ⱦɥɹ ɧɟɟ Q, ɨɱɟɜɢɞɧɨ, ɪɚɜɧɨ ɧɭɥɸ. |
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ɉɪɟɞɩɨɥɨɠɢɦ, ɱɬɨ ɷɬɚ ɫɨɜɨɤɭɩɧɨɫɬɶ ɦɨɠɟɬ ɛɵɬɶ ɩɨ ɤɚɤɨɦɭ-ɥɢɛɨ ɩɪɢɡɧɚɤɭ (ɧɚɩɪɢɦɟɪ, |
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ɩɨ ɩɨɥɭ) ɪɚɡɛɢɬɚ ɧɚ 2 ɫɨɜɨɤɭɩɧɨɫɬɢ: |
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ɚ |
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ɛ |
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X |
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Y |
N(xi) |
X |
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Y |
N(xi) |
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y1 |
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y2 |
y1 |
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y2 |
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x1 |
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100 |
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50 |
150 |
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x1 |
200 |
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250 |
450 |
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x2 |
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50 |
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150 |
200 |
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x2 |
150 |
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50 |
200 |
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N(yj) |
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150 |
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200 |
350 |
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N(yj) |
350 |
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300 |
650 |
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Ⱦɥɹ ɩɟɪɜɨɣ Q=+0,71, ɞɥɹ ɜɬɨɪɨɣ Q= — 0,58.
Ɍɚɤɢɦ ɨɛɪɚɡɨɦ, ɞɥɹ ɨɞɧɨɣ ɩɨɞɫɨɜɨɤɭɩɧɨɫɬɢ (ɧɚɩɪɢɦɟɪ, ɞɥɹ ɦɭɠɱɢɧ) ɫɜɹɡɶ ɦɟɠɞɭ ɩɪɢɡɧɚɤɚɦɢ X ɢ Y ɩɨɥɨɠɢɬɟɥɶɧɚɹ, ɚ ɞɥɹ ɞɪɭɝɨɣ (ɞɥɹ ɠɟɧɳɢɧ) — ɨɬɪɢɰɚɬɟɥɶɧɚɹ.
ɗɬɨɬ ɩɪɢɦɟɪ ɮɨɪɦɚɥɶɧɨ ɢɥɥɸɫɬɪɢɪɭɟɬ ɫɥɭɱɚɣ, ɤɨɝɞɚ ɫɜɹɡɶ ɨɤɚɡɚɥɚɫɶ ɫɤɪɵɬɨɣ. ɇɟɫɥɨɠɧɨ ɫɤɨɧɫɬɪɭɢɪɨɜɚɬɶ ɩɪɢɦɟɪ, ɤɨɝɞɚ ɜɨɡɧɢɤɚɸɬ ɤɚɠɭɳɢɟɫɹ ɫɜɹɡɢ. Ⱦɟɥɨ ɡɞɟɫɶ,
ɤɨɧɟɱɧɨ, ɧɟ ɜ «ɩɨɞɝɨɧɤɟ» ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɯ ɬɚɛɥɢɰ, ɚ ɜ ɬɨɦ, ɱɬɨ ɩɨɞɨɛɧɵɟ ɷɮɮɟɤɬɵ ɦɨɝɭɬ
ɢɦɟɬɶ ɦɟɫɬɨ ɜ ɪɟɚɥɶɧɨɣ ɫɢɬɭɚɰɢɢ. Ʉɚɤ ɢɡɛɟɠɚɬɶ ɢɯ?
Ⱦɟɬɚɥɶɧɵɟ ɪɟɤɨɦɟɧɞɚɰɢɢ ɞɚɜɚɬɶ ɬɪɭɞɧɨ, ɧɨ ɜɚɠɧɨ, ɱɬɨɛɵ ɫɨɰɢɨɥɨɝ ɧɟ ɩɪɢɦɟɧɹɥ ɤɨɷɮɮɢɰɢɟɧɬɵ ɛɟɡɞɭɦɧɨ. ɇɭɠɧɨ ɨɫɦɵɫɥɢɜɚɬɶ ɢɡɭɱɚɟɦɭɸ ɫɢɬɭɚɰɢɸ, ɭɞɟɥɹɬɶ ɛɨɥɶɲɨɟ ɜɧɢɦɚɧɢɟ ɨɞɧɨɪɨɞɧɨɫɬɢ ɢɡɭɱɚɟɦɵɯ ɫɨɰɢɚɥɶɧɵɯ ɨɛɳɧɨɫɬɟɣ (ɷɬɨ ɧɟ ɨɡɧɚɱɚɟɬ, ɤɨɧɟɱɧɨ, ɱɬɨ ɧɟɥɶɡɹ ɜɵɞɟɥɹɬɶ ɢ ɢɫɫɥɟɞɨɜɚɬɶ ɩɚɪɚɥɥɟɥɶɧɨ ɪɚɡɧɨɪɨɞɧɵɟ ɝɪɭɩɩɵ).
ɂ, ɧɚɤɨɧɟɰ, ɨ ɫɜɹɡɹɯ ɤɨɷɮɮɢɰɢɟɧɬɨɜ Q ɢ ĭ ɫ ij ɢ ɋ.
ɋ ɭɱɟɬɨɦ (II,1,2) |
ɢ (I,3,2) |
ɥɟɝɤɨ ɜɢɞɟɬɶ, ɱɬɨ ɞɥɹ ɬɚɛɥɢɰ 2 × 2: )2 |
F2 |
. ɋ ɞɪɭɝɨɣ |
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N |
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ɫɬɨɪɨɧɵ, ɤɚɤ ɦɵ ɜɢɞɟɥɢ, |
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[92] |
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ɞɥɹ ɬɚɛɥɢɰ k × l: |
M2 |
F2 |
, |
ɬ.ɟ. ij2 ɹɜɥɹɟɬɫɹ ɨɛɨɛɳɟɧɢɟɦ ĭ ɧɚ ɫɥɭɱɚɣ ɤɨɪɪɟɥɹɰɢɨɧɧɵɯ |
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ɬɚɛɥɢɰ ɨɛɳɟɝɨ ɜɢɞɚ.
ȼ ɤɚɱɟɫɬɜɟ ɫɜɨɟɨɛɪɚɡɧɨɝɨ ɨɛɨɛɳɟɧɢɹ Q ɢ ĭ ɦɨɠɧɨ ɪɚɫɫɦɚɬɪɢɜɚɬɶ ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɫɪɟɞɧɟɣ ɤɜɚɞɪɚɬɚ ɱɟɫɤɨɣ ɫɨɩɪɹɠɟɧɧɨɫɬɢ ɋ.
Ɉɫɜɹɡɢ ĭ ɤɨɷɮɮɢɰɢɟɧɬɚ ɫ ɤɨɷɮɮɢɰɢɟɧɬɨɦ Ʉɟɧɞɷɥɚ ɫɦ. ɜ ɤɨɧɰɟ §6 ɷɬɨɣ ɝɥɚɜɵ.
4.Ʉɨɷɮɮɢɰɢɟɧɬ ɪɚɧɝɨɜɨɣ ɤɨɪɪɟɥɹɰɢɢ ɋɩɢɪɦɟɧɚ
62
Ɋɚɫɫɦɨɬɪɟɧɧɵɟ ɪɚɧɟɟ ɦɟɪɵ ɛɚɡɢɪɭɸɬɫɹ, ɤɚɤ ɨɬɦɟɱɚɥɨɫɶ, ɧɚ ɩɪɢɧɰɢɩɟ ɫɨɜɦɟɫɬɧɨɝɨ ɩɨɹɜɥɟɧɢɹ ɫɨɛɵɬɢɣ. Ɉɧɢ ɩɪɢɝɨɞɧɵ ɞɥɹ ɥɸɛɵɯ ɩɪɢɡɧɚɤɨɜ — ɦɟɬɪɢɱɟɫɤɢɯ, ɩɨɪɹɞɤɨɜɵɯ ɢ ɞɚɠɟ ɧɨɦɢɧɚɥɶɧɵɯ.
Ⱦɥɹ ɦɟɬɪɢɱɟɫɤɢɯ ɢ ɩɨɪɹɞɤɨɜɵɯ ɩɪɢɡɧɚɤɨɜ ɦɨɝɭɬ ɢɫɩɨɥɶɡɨɜɚɬɶɫɹ ɦɟɪɵ, ɨɫɧɨɜɚɧɧɵɟ ɧɚ ɩɪɢɧɰɢɩɟ ɤɨɜɚɪɢɚɰɢɢ. Ƚɨɜɨɪɹɬ, ɱɬɨ ɩɟɪɟɦɟɧɧɵɟ ɤɨɜɚɪɢɚɧɬɧɵ, ɟɫɥɢ ɜɚɪɢɚɰɢɢ ɨɞɧɨɣ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɜɚɪɢɚɰɢɹɦ ɞɪɭɝɨɣ. ɉɪɢɧɰɢɩ ɤɨɜɚɪɢɚɰɢɢ, ɞɪɭɝɢɦɢ ɫɥɨɜɚɦɢ, ɨɫɧɨɜɚɧ ɧɚ ɢɡɭɱɟɧɢɢ ɫɨɜɦɟɫɬɧɵɯ ɢɡɦɟɧɟɧɢɣ ɜ ɡɧɚɱɟɧɢɹɯ ɩɪɢɡɧɚɤɨɜ. əɫɧɨ, ɱɬɨ ɟɝɨ ɦɨɠɧɨ ɢɫɩɨɥɶɡɨɜɚɬɶ ɞɥɹ ɤɨɥɢɱɟɫɬɜɟɧɧɵɯ ɞɚɧɧɵɯ, ɨɞɧɚɤɨ ɫɨɰɢɚɥɶɧɵɟ ɩɪɢɡɧɚɤɢ ɡɚɱɚɫɬɭɸ ɞɨɩɭɫɤɚɸɬ ɬɨɥɶɤɨ ɭɩɨɪɹɞɨɱɟɧɢɟ. ɇɚɩɪɢɦɟɪ, ɨɪɢɟɧɬɚɰɢɢ, ɨɰɟɧɤɢ, ɭɞɨɜɥɟɬɜɨɪɟɧɧɨɫɬɢ, ɹɜɥɹɸɳɢɟɫɹ ɫɨɛɫɬɜɟɧɧɨ ɫɨɰɢɨɥɨɝɢɱɟɫɤɢɦɢ ɩɟɪɟɦɟɧɧɵɦɢ, ɩɨ ɫɭɳɟɫɬɜɭ ɢɡɦɟɪɹɸɬɫɹ ɫ ɩɨɦɨɳɶɸ ɲɤɚɥ ɩɨɪɹɞɤɚ: ɫɨɨɬɜɟɬɫɬɜɭɸɳɢɟ ɷɦɩɢɪɢɱɟɫɤɢɟ ɩɪɨɰɟɞɭɪɵ, ɤɚɤ ɦɵ ɜɢɞɟɥɢ, ɞɚɸɬ ɜɨɡɦɨɠɧɨɫɬɶ ɫɤɚɡɚɬɶ, ɱɬɨ ɢɧɞɢɜɢɞ Ⱥ ɛɨɥɟɟ ɭɞɨɜɥɟɬɜɨɪɟɧ, ɱɟɦ ȼ, ɫɜɨɟɣ ɫɩɟɰɢɚɥɶɧɨɫɬɶɸ, ɧɚɩɪɢɦɟɪ, ɧɨ ɧɟ ɩɨɡɜɨɥɹɸɬ ɫɤɚɡɚɬɶ ɧɚ ɫɤɨɥɶɤɨ (ɬɟɦ ɛɨɥɟɟ — ɜɨ ɫɤɨɥɶɤɨ ɪɚɡ) ɛɨɥɶɲɟ.
ȿɫɥɢ ɫɨɜɨɤɭɩɧɨɫɬɶ ɭɩɨɪɹɞɨɱɟɧɚ ɩɨ ɞɜɭɦ (ɢɥɢ ɛɨɥɟɟ) ɩɪɢɡɧɚɤɚɦ ɢ ɢɡɦɟɧɟɧɢɸ ɨɞɧɨɝɨ ɩɪɢɡɧɚɤɚ ɫɨɨɬɜɟɬɫɬɜɭɟɬ ɢɡɦɟɧɟɧɢɟ ɞɪɭɝɨɝɨ, ɬɨ ɝɨɜɨɪɹɬ ɨ ɧɚɥɢɱɢɢ ɤɨɪɪɟɥɹɰɢɢ ɦɟɠɞɭ ɧɢɦɢ. ɑɟɦ ɦɨɠɧɨ ɢɡɦɟɪɢɬɶ ɷɬɭ ɤɨɪɪɟɥɹɰɢɸ?
ɋɩɢɪɦɟɧɨɜɫɤɢɣ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɪɚɧɝɨɜ. Ⱦɨɩɭɫɬɢɦ, ɱɬɨ N ɢɧɞɢɜɢɞɨɜ ɦɨɝɭɬ ɛɵɬɶ ɭɩɨɪɹɞɨɱɟɧɵ ɤɚɤ ɩɨ ɩɪɢɡɧɚɤɭ X, ɬɚɤ ɢ ɩɨ ɩɪɢɡɧɚɤɭ Y. ɉɭɫɬɶ Ri(x) — ɪɚɧɝ i-ɝo ɢɧɞɢɜɢɞɚ ɩɨ ɩɪɢɡɧɚɤɭ X ( i 1,N ) , ɚ Ri(y) — ɩɨ Y. Ɇɟɪɨɣ ɧɟɫɨɜɩɚɞɟɧɢɹ ɢɯ ɹɜɥɹɟɬɫɹ ɜɟɥɢɱɢɧɚ di=Ri(x)— Ri(y). ȼɨ ɢɡɛɟɠɚɧɢɟ ɷɮɮɟɤɬɚ ɤɨɦɩɟɧɫɚɰɢɢ, ɤɚɤ ɢ ɪɚɧɟɟ, ɩɪɢ ɩɟɪɟɯɨɞɟ ɤ ɩɨɥɧɨɣ ɦɟɪɟ ɜɨɡɜɟɞɟɦ
N
di, ɜ ɤɜɚɞɪɚɬ ɢ ɫɥɨɠɢɦ, ɬ.ɟ. ɪɚɫɫɦɨɬɪɢɦ ¦di2 .
i 1
[93]
ɉɨɬɪɟɛɭɟɦ ɞɚɥɟɟ, ɱɬɨɛɵ: 1) ɢɫɤɨɦɵɣ ɤɨɷɮɮɢɰɢɟɧɬ ɤɨɪɪɟɥɹɰɢɢ ɪɚɧɝɨɜ ɨɛɪɚɳɚɥɫɹ ɜ +1,
ɟɫɥɢ ɜɫɟ ɪɚɧɝɢ ɫɨɜɩɚɞɚɸɬ, ɢ 2) ɜ (—1), ɟɫɥɢ ɪɚɧɝɨɜɵɟ ɪɹɞɵ ɢɦɟɸɬ ɨɛɪɚɬɧɨɟ ɧɚɩɪɚɜɥɟɧɢɟ
(ɬɚɤ, ɞɥɹ N=5, Ri(x)=1, 2, 3, 4, 5, ɚ Ri(y)=5, 4, 3, 2, 1).
N
ɋɬɚɧɟɦ ɢɫɤɚɬɶ ɷɬɨɬ ɤɨɷɮɮɢɰɢɟɧɬ ɜ ɜɢɞɟ 1 f ¦d i2
i 1
(ɜɟɥɢɱɢɧɭ f ɦɵ ɧɚɣɞɟɦ ɱɭɬɶ ɩɨɡɞɧɟɟ), ɬɨɝɞɚ ɩɟɪɜɨɟ ɬɪɟɛɨɜɚɧɢɟ ɜɵɩɨɥɧɹɟɬɫɹ
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ɚɜɬɨɦɚɬɢɱɟɫɤɢ: ɟɫɥɢ ɪɚɧɝɨɜɵɟ ɪɹɞɵ ɢɞɟɧɬɢɱɧɵ, ɬɨ ¦di2 |
0 |
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i 1 |
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ɢ ɤɨɷɮɮɢɰɢɟɧɬ ɪɚɜɟɧ 1. ȼɵɛɟɪɟɦ f ɬɚɤ, ɱɬɨɛɵ ɭɞɨɜɥɟɬɜɨɪɢɬɶ ɜɬɨɪɨɦɭ ɬɪɟɛɨɜɚɧɢɸ. |
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Ⱦɨɩɭɫɬɢɦ, ɫɩɟɪɜɚ, ɱɬɨ N ɱɟɬɧɨ. ɇɚɩɪɢɦɟɪ, ɞɥɹ N=6 ɢɦɟɟɦ: |
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Ri(x) |
1 |
2 |
3 |
4 |
5 |
6 |
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Ri(y) |
6 |
5 |
4 |
3 |
2 |
1 |
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di2 |
52 |
32 |
12 |
12 |
32 |
52 |
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ɉɪɢ N=2k: |
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Ri(x) |
1 |
2 |
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… |
k–1 |
k |
k+1 |
k+2 |
… |
2k–1 |
2k |
Ri(y) |
2k |
2k–1 |
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… |
k+2 |
k+1 |
k |
k–1 |
… |
2 |
1 |
di2 |
(2k–1)2 |
(2k–3)2 |
… |
32 |
12 |
12 |
32 |
… |
(2k–3)2 |
(2k–1)2 |
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¦di2 2>12 32 ... ( 2k 1) 2 @ 1 k( 4k 2 1)
3
1
(ɫɦ. ɉɪɢɥɨɠɟɧɢɟ 2), ɫɥɟɞɨɜɚɬɟɥɶɧɨ, ¦di2 6 N( N 2 1)
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