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# Measures of central tendency

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### Measures of central tendency

1. 1. MEASURES OF CENTRAL TENDENCY<br /><ul><li>Ukuran Nilai Pusat
2. 2. Ukuran Pemusatan
3. 3. Ukuran Kecenderungan Sentral
4. 4. Ukuran Sentral Tendensi
5. 5. Ukuran Tendensi Sentral
6. 6. Ukuran Gejala Pusat</li></ul>Ia Kurnia, Drs., M.Pd.<br />1<br />
7. 7. Measures of Central Tendency<br />Setelah mengikuti sesi perkuliahan ini, mahasiswa diharapkan mampu:<br /><ul><li>Menjelaskan pengertian “measures of central tendency”
8. 8. Membedakan jenis-jenis “measures of central tendency”
9. 9. Menghitung jenis-jenis “measures of central tendency” dan menafsirkan hasilnya. </li></ul>Ia Kurnia, Drs., M.Pd.<br />2<br />
10. 10. Measures of Central Tendency<br />Nilai Tunggal yang representatif bagi seluruh nilai data.<br />Ia Kurnia, Drs., M.Pd.<br />3<br />
11. 11. Measures of Central Tendency<br />Jenis:<br />I Mathematical Averages<br />Arithmetic Mean<br />Geometric Mean<br />Harmonic Mean<br />Quadratic Mean (Root Mean Square)<br />II. Positional Averages<br />Median<br />Mode<br />Quartiles<br />Deciles <br />Percentiles<br />Ia Kurnia, Drs., M.Pd.<br />4<br />4<br />
12. 12. ARITHMETIC MEAN<br />Grouped Data<br />Ungrouped Data<br />Ia Kurnia, Drs., M.Pd.<br />5<br />
13. 13. ARITHMETIC MEAN<br />Ia Kurnia, Drs., M.Pd.<br />6<br />
14. 14. ARITHMETIC MEAN<br />Merits of Arithmetic Mean<br /><ul><li>The Arithmetic Mean is the most widely used measure of central tendency,
15. 15. Its definition is clear and precise. It corresponds to the centre of gravity of the observation
16. 16. It is simple to understand and easy to compute.
17. 17. It use each and every item in the data
18. 18. It has a determinate value and is rigidly defined
19. 19. It can be subjected to further algebraic treatment and advanced statistical theory is based on it
20. 20. It can be found event if only the total of values is known and the individual values are not known
21. 21. It provides a good standard of comparison since extreme values can cancel each other when the number of observation is large</li></ul>Ia Kurnia, Drs., M.Pd.<br />7<br />“The Arithmetic Mean is obtained by dividing the sum of values of observation by the number of observation”<br />
22. 22. ARITHMETIC MEAN<br />Demerits of Arithmetic Mean<br /><ul><li>It may be sometimes greatly influenced by unrepresentative values. The mean of 49, 50, 51 is 50. But the mean of 49, 50, 51, 250 is 100. In such cases the representative character of the mean is lost
23. 23. It gives greater importance to larger and less importance to smaller values. It has an upward bias
24. 24. It cannot be calculated if one more items in the data are missing
25. 25. It cannot be located by inspection (like the mode and median)
26. 26. It may conceal facts and may lead to distorted conclusions.</li></ul>Thus, if the performance of two students in three terms is given by<br />(a) 30% 40% 50% marks<br />(b) 50% 40% 30% marks<br />Although the mean are equal we cannot know from the means alone that one students is showing improved results, and the other deteriorating. Similarly inequalities of incomes are concealed by the per capita incomes and the average result of a collage may tell nothing about the general performance of the students. <br />Ia Kurnia, Drs., M.Pd.<br />8<br />4<br />
27. 27. Ia Kurnia, Drs., M.Pd.<br />9<br />GEOMETRIC MEAN<br />Ungrouped Data<br />(1)<br />Untuk data yang perubahannya mengikuti atau dianggap mengikuti aturan-aturan tertentu, misalnya pertambahan penduduk, perubahan modal/tabungan, pertumbuhan bakteri dll.<br />Pt = Keadaan pada akhir periode<br />Po = Keadaan awal<br />X = rata-rata<br />t = jangka waktu atau lamanya periode<br />(2)<br />
28. 28. GEOMETRICMEAN<br />Ia Kurnia, Drs., M.Pd.<br />10<br />Grouped Data<br />Notes<br />Fi = Frekuensi Kelas<br />Mdi = Mid Point Kelas<br />
29. 29. Ia Kurnia, Drs., M.Pd.<br />11<br />GEOMETRICMEAN<br />
30. 30. GEOMETRICMEAN<br />Merits of Geometric Mean<br />Most of the properties and merits of GM follow the same lines as those of AM.<br /><ul><li>The GM takes into account all the items in a series and condenses them into one representative value
31. 31. It has a downward bias. It gives more weight to small values than to large values
32. 32. It is determinate. For the same data there cannot be two geometric mean
33. 33. It balances to ratios of the values on either side of the data. It is ideally suited to average rate of change such as index numbers and ratios between measures and percentages.
34. 34. It is amenable to algebraic manipulations like the AM.</li></ul>Demerits of Geometric Mean<br /><ul><li>It is difficult to understand and to compute
35. 35. It is determined for positive values and cannot be used for negative values or zero.</li></ul>Ia Kurnia, Drs., M.Pd.<br />12<br />4<br />
36. 36. MEDIAN (Me)<br />Adalah sebuah bilangan yang bersifat bahwa setengah dari data (setelah diurutkan) akan lebih kecil dari atau sama besar dengan bilangan itu, dan setengahnya lagi akan lebih besardari atau sama dengan bilangan tersebut.<br />Ungrouped Data<br /><ul><li> Data diurutkan secara ascending
37. 37. Median (Me) = Data ke</li></ul> n = Banyak data<br />Grouped Data<br />Ia Kurnia, Drs., M.Pd.<br />13<br />LCBme = Lower Class Boundary Median terletak<br />Ci = Panjang Class Interval<br />FC(me-1) = Frekuensi Kumulatif sblm kelas median<br />Fme = Frekuensi Kelas Median<br />n = Jumlah Data Observasi <br />
38. 38. MEDIAN (Me)<br />Merits of the Median<br /><ul><li>It can be easily understood and its composition is simple
39. 39. It can be computed even for incomplete data. It is concerned only with a few central observation
40. 40. It balances the number of items in a distribution. It is useful in describing scores, ratios and grades
41. 41. It is more useful in the case of skewed distributions like those of incomes and prices
42. 42. It can be used for qualitative data
43. 43. In the case of open end classes the median can be calculated but the Mean cannot
44. 44. It can be easily determined graphically (see Ogives)</li></ul>Ia Kurnia, Drs., M.Pd.<br />14<br />
45. 45. MEDIAN (Me)<br />Demerits of the Median<br /><ul><li>Rearrangement of data may be necessary to compute the median
46. 46. The median is not capable of algebraic manipulation. As such it is not used in advanced studies
47. 47. The empirical formula for the median based on interpolation may not always give correct results
48. 48. It may ignore significant extreme values
49. 49. Weighting cannot be used in the case of the median. The scope of operations is thus narrowed
50. 50. It cannot be computed as exactly as the mean </li></ul>Ia Kurnia, Drs., M.Pd.<br />15<br />3<br />
51. 51. MODE<br />Modus Nilai data yang paling sering muncul atau nilai data yang mempunyai frekuensi terbanyak.<br />Ungrouped Data<br />Mencari nilai data dengan kemunculan terbanyak<br />Grouped Data<br />LCBmo = Lower Class Boundary, modus terletak<br />Ci = Panjan Interval Kelas<br />Fmo = Frekuensi kelas modus<br />Fmo-1 = Frekuensi sebelum kelas modus<br />Fmo+1 = Frekuensi setelah kelas modus<br />Ia Kurnia, Drs., M.Pd.<br />16<br />
52. 52. MODE <br />Merits and use of the Mode<br /><ul><li>It is simply defined and computed.
53. 53. It is the most popular average in the sense that is the one that most people use without being aware use without being average of it e.g. when they speak of the average number of car accident, bus breakdown etc.
54. 54. Extreme value have no effect on the mode and it can be calculated when complete data are not available
55. 55. It is the most typical in the sense that it denotes the most probable value in the series</li></ul>Demerits of the Mode<br /><ul><li>In the case of bimodal or multimodal distributions, it is impossible to pinpoint any one value as the mode
56. 56. It is not rigidly defined and thus cannot be called an ideal average
57. 57. It is often indeterminate when the distribution is highly irregular
58. 58. It is not based on all the observation in the series and hence is not an ideal measure of central tendency
59. 59. It is not amenable to algebraic manipulation </li></ul>Ia Kurnia, Drs., M.Pd.<br />17<br />2<br />
60. 60. HUBUNGAN Xbar, Me, dan Mo<br />Ia Kurnia, Drs., M.Pd.<br />18<br />Xbar = Me = Mo<br /> Mo Me Xbar<br />Symmetric/Normal Curve<br />Populasi/sampel dengan bentuk kurva ini, nilai Xbar, Me dan Mo akan sama.<br />Skewed Negative Curve<br />Populasi/sampel dengan bentuk kurva ini, nilai Xbar> Me > Mo<br />
61. 61. HUBUNGAN Xbar,, Me dan Mo<br />Ia Kurnia, Drs., M.Pd.<br />19<br /> XbarMe Mo<br />2<br />Skewed Positive Curve<br />Populasi/sampel dengan bentuk kurva ini, nilai Xbar < Me < Mo<br />Untuk Symmetric Curve, berlaku hubungan:<br />Xbar – Mo = 5/4(Xbar – Me)<br />Untuk Skewed Positive/Negative<br />Xbar – Mo = 3(Xbar – Me)<br />
62. 62. CHOICE OF AN “AVERAGE”<br />Is there such a thing as an Ideal Average? The answer is ‘probability not’ because no such ideal average has yet been computed. But what do we mean by ideal? It is generally agreed that the title should be given to that average which possesses that following characteristics:<br />It is rigidly defined and easily computed<br />It is simple enough to be understood by the layman<br />It takes into account all item in the series and gives them equal weights. It is not unduly influenced by a few extreme values<br />Its values are not greatly affected by sampling fluctuations<br />It is amenable to algebraic manipulations.<br />From the point listed above it obvious that all measures of central tendency discussed in the proceeding pages posses some of these characteristic and the arithmetic mean satisfies them to a greater degree than any other measure. The arithmetic mean is rigidly defined , is invariably determinate and quite easily computed. It is in common use, takes into account all items and its value is not much influenced by sampling fluctuations. Most important, it is amenable to algebraic manipulations. Its only important drawback is that it is influenced by a few high values in values in extremes. For all that is the average most often opted for when a choice has to be made. <br />Ia Kurnia, Drs., M.Pd.<br />20<br />
63. 63. CHOICE OF AN “AVERAGE”<br />There are three factors that have to be kept in mind when deciding what average to use:<br />The purpose for which the average is being used<br />The nature, characteristics and properties of the average<br />The nature and characteristics of the data: in particular, the degree of homogeneity in the data. <br />Ia Kurnia, Drs., M.Pd.<br />21<br />2<br />
64. 64. QUARTILE, DECILE, PERCENTILE<br />QUARTILE/KUARSIL<br />Bilangan-bilangan yang membagi sekelompok data menjadi empat bagian yang sama. Bilangan itu: Q1, Q2, Q3<br />Q1 : 25% dari data akan lebih kecil atau sama dengan bilangan itu.<br />Q2 : 50% dari data akan lebih kecil atau sama dengan bilangan itu <br />Q3 : 75% dari data akan lebih kecil atau sama dengan bilangan itu<br />Ungrouped Data<br />Ia Kurnia, Drs., M.Pd.<br />22<br />
65. 65. QUARTILE, DECILE, PERCENTILE<br />Grouped Data<br />Ia Kurnia, Drs., M.Pd.<br />23<br />DECILE/DESIL<br />Bilangan-bilangan yang membagi sekelompok data menjadi sepuluh bagian yang sama. Bilangan itu: D1, D2, D3,….,D9<br />D1 : 10% dari data akan lebih kecil atau sama dengan bilangan itu.<br />D2 : 30% dari data akan lebih kecil atau sama dengan bilangan itu <br />D9 : 90% dari data akan lebih kecil atau sama dengan bilangan itu<br />
66. 66. QUARTILE, DECILE, PERCENTILE<br />Ungrouped Data<br />Grouped Data<br />PERCENTILE/PERSENTIL<br />Bilangan-bilangan yang membagi sekelompok data menjadi seratus bagian yang sama. Bilangan itu: <br /> P1, P2, P3,….,P99<br />Ia Kurnia, Drs., M.Pd.<br />24<br />
67. 67. QUARTILE, DECILE, PERCENTILE<br />Ungrouped Data<br />Grouped Data<br />Ia Kurnia, Drs., M.Pd.<br />25<br />
68. 68. QUARTILE, DECILE, PERCENTILE<br />Berikut ini adalah data harga Per Lembar Saham pada 20 perusahaan pilihan di Bursa Efek Jakarta<br />Ia Kurnia, Drs., M.Pd.<br />26<br />Contoh<br />
69. 69. Laba bersih setahun yang diperoleh 120 perusahaan di Kota Damai<br />Ia Kurnia, Drs., M.Pd.<br />27<br />QUARTILE, DECILE, PERCENTILE<br />6<br />Contoh<br />
70. 70. Ia Kurnia, Drs., M.Pd.<br />28<br />