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Measures of central tendency

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Presentasi ke-2 Statistika Untuk Ekonomi …

Presentasi ke-2 Statistika Untuk Ekonomi
STIE Muhammadiyah Bandung

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

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