5. Ukuran Sebaran ( keragaman )

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Kuswanto 2013

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5. Ukuran Sebaran ( keragaman ). Kuswanto 2013. Ukuran keragaman. Dari tiga ukuran pemusatan, belum dapat memberikan deskripsi yang lengkap bagi suatu data. Perlu juga diketahui seberapa jauh pengamatan-pengamatan tersebut menyebar dari rata-ratanya. - PowerPoint PPT Presentation

Transcript of 5. Ukuran Sebaran ( keragaman )

Page 1: 5.  Ukuran Sebaran  ( keragaman )

Kuswanto 2013

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Ukuran keragaman Dari tiga ukuran pemusatan, belum dapat memberikan

deskripsi yang lengkap bagi suatu data. Perlu juga diketahui seberapa jauh pengamatan-

pengamatan tersebut menyebar dari rata-ratanya. Ada kemungkinan diperoleh rata-rata dan median yang

sama, namun berbeda keragamannya. Beberapa ukuran keragaman yang sering kita temui

adalah range (rentang=kisaran=wilayah), simpangan (deviasi), varian (ragam), simpangan baku (standar deviasi) dan koefisien keragaman.

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Measures of Dispersion and Variability

These are measurements of how spread the data is around the center of the distribution

f

X

f

X

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1.Range Kisaran = Rentangdifference between lowest and highest numbers

Place numbers in order of magnitude,then range = Xn - X1.

Range = 5 - 2 = 32

2345

= X1

= X2

= X3

= X4

= X5

Problem - no information about how clustered the data is

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2. SIMPANGAN (deviation)

You could express dispersion in terms of deviation from the mean, however, a sum of deviations from the mean will always = 0.

i.e. (Xi - X) = 0

So, take an absolute value to avoid this

Problem – the more numbers in the data set, the higher the SS

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Contoh Misal, jumlah buku tulis yang dibawa 5 mahasiswa adalah 3, 5, 7, 7,

8. Rerata (mean) data tersebut adalah 30/5 = 6. Simpangan dihitung dengan mengurangi setiap nilai pengamatan dengan reratanya

No. Nilai observasi

Simpangan(x - x)

Simpangan kuadrat (x - x)2

1 3 3-6 = -3 92 5 5-6 = -1 13 7 7-6 = 1 14 7 7-6 = 1 15 8 8-6 = 2 4

Jumlah 0 16

Agar nilainya tidak negatip, dapat di kuadratkan yang kemudian

disebut simpangan kuadrat. Simpangan kuadrat bermanfaat untuk penghitungan varian.

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Sample mean deviation = | Xi - X |n

Essentially the average deviation from the mean

3. Simpangan Rerata (Mean Deviation)

No. Nilai observasi Simpangan(x - x)

1 3 3-6 = -32 5 5-6 = -13 7 7-6 = 14 7 7-6 = 15 8 8-6 = 2

Jumlah 0

Simpangan rerata(x - x)/n

(3-6)/5 = -3/5(5-6)/5 = -1/5(7-6)/5 = 1/5(7-6)/5 = 1/5(8-6)/5 = 2/5

0

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4. Sum of square

Sample SS = (Xi - X)2 =

SS is much more common than mean deviation

Another way to get around the problem of zero sums is to square the deviations. Known as sum of squares or SS

Xi2 - (Xi)2/n

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Example

22345

= X1

= X2

= X3

= X4

= X5

X = 3.2

Sample SS = (Xi - X)2

SS = (2 - 3.2)2 + (2 - 3.2)2 + (3 - 3.2)2 + (4 - 3.2)2 + (5 -3.2)2

= 1.44 + 1.44 + 0.04 + 0.64 + 3.24

= 6.8

Problem – the more numbers in the data set, the higher the SS

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5. VARIAN (RAGAM) Dalam prakteknya, simpangan jarang digunakan

karena sulit dimanipulasi secara matematis. Sebagai gantinya diperlukan kuadrat semua

simpangan tersebut kemudian dibagi derajad bebas n-1, dan disebut dengan varian (ragam)

Digunakan pembagi n-1 agar menjadi penduga tak bias.

Ragam populasi dilambangkan dengan σ², sedang ragam contoh dilambangkan dengan s2,

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The mean SS is known as the variance

Population Variance (2 ):

2 = (Xi - )2

N

This is just SSN

Problem - units end up squared

Our best estimate of 2 is sample variance (s2):

S2 = (Xi - X)2 n - 1 Note : divide by n-1

known as degrees of freedom Xi2 - (Xi)2/nn - 1

=

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Mengapa (n-1) disebut derajad bebas (kebebasan)?

Perhatikan ilustrasi berikut. Apabila seseorang hendak mengangkat 100 kg beras

dari lantai 1 ke lantai 3 dan ia harus mengangkat maksimal sebanyak 5 kali, maka orang tersebut dapat memilih menyelesaikannya dalam 2 kali angkat, 3 kali atau sampai (n-1) kali.

Sampai dengan 4 (n-1) kali, orang tersebut bebas memilih berapa kg yang diangkat ke lantai 3.

Namun pada angkatan terakhir (1 kali), mau tidak mau, orang tersebut harus mengangkat semua beras yang tersisa. Artinya kebebasan memilih jumlah yang diangkat hanya (n-1) kali.

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6. Standar Deviasi Penggunaan ragam untuk mengukur keragaman,

diperoleh satuan kuadrat dari satuan semula. Apabila yang dihitung keragamannya adalah bobot buah

melon dengan satuan kg, maka ragamnya akan mempunyai satuan kg².

Apabila yang diukur keragamannya adalah jumlah petani dengan satuan orang, maka ragamnya akan mempunyai satuan orang²??. Tentu saja hal ini sangat tidak logis.

Agar diperoleh satuan yang sama dengan satuan asalnya, maka varian tersebut diakarkan. Akar dari ragam disebut simpangan baku (s) atau dikenal dengan standar deviasi

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Standard Deviation (Standar Deviasi)=> square root of variance

= (Xi - )2

N

For a population:

For a sample:

s = (Xi - X )2

n - 1

= 2

s = s2

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Contoh

22345

= X1

= X2

= X3

= X4

= X5

X = 3.2

s = (Xi - X )2

n - 1

s = (2 - 3.2)2 + (2 - 3.2)2 + (3 - 3.2)2 + (4 - 3.2)2 + (5 -3.2)2

5 - 1

= 1.44 + 1.44 + 0.04 + 0.64 + 3.24

= 1.3044

Hitung standar deviasi

Varian adalah kuadrat dari standar deviasi. Berapa nilainya??

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7. Coefficient of Variation = Koefisien Keragaman = KK (V or sometimes CV):

CV =sX

Variance (s2) and standard deviation (s) have magnitudes that are dependent on the magnitudes of the data.

The coefficient of variation is a relative measure, so variability of different sets of data may be compared (stdev relative to the mean)

Note that there are no units – emphasizes that it is a relative measure

Sometimes expressed as a %

X 100%

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Example:

22345

= X1

= X2

= X3

= X4

= X5

s = 1.304 g

CV =sX

X = 3.2 g

CV = 1.304 g3.2 g

CV = 0.4075

orCV = 40.75%

(X 100%)

Attention there is not any UNIT, or %

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8. The Normal Distribution (Distribusi Normal) :

There is an equation which describes the height of the normal curve in relation to its standard dev ()

X 2

3

2

3

68.27%

95.44%

99.73%

f

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ƒ

-3 -2 -1 0 1 2 3 4

μ = 0

Normal distribution with σ = 1, with varying means

μ = 1 μ = 2

5

If you get difficulties to keep this term, read statistics books

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ƒ

-4 -3 -2 -1 0 1 2 3-5 4 5

σ = 1

σ = 1.5

σ = 2

Normal distribution with μ = 0, with varying standard deviations

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9. Symmetry and KurtosisSymmetry means that the population is equally distributed around the mean i.e. the curve to the right side of the mean is a mirror image of the curve to the left side

ƒ

Mean, median and mode

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Data may be positively skewed (skewed to the right)

Symmetry ƒ

ƒ

Or negatively skewed (skewed to the left)

So direction of skew refers to the direction of longer tail

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Symmetry

ƒ

mode

medianmean

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ƒKurtosis refers to how flat or peaked a curve is (sometimes referred to as peakedness or tailedness)

The normal curve is known as mesokurtic

ƒ

A more peaked curve is known as leptokurtic

A flatter curve is known as platykurtic

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Latihan dan diskusi1. Banyaknya buah pisang yang tersengat hama dari 16 tanaman

adalah 4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, dan 15. Dengan menganggap data tersebut sebagai contoh, hitunglah varian, simpangan baku dan koefisien keragamannya. Statistik mana yang paling tepat untuk menggambarkan keragaman data tersebut?

2. To study how first-grade students utilize their time when assigned to a math task, researcher observes 24 students and records their time off task out of 20 minutes. Times off task (minutes) : 4, 0, 2, 2, 4, 1, 4, 6, 9, 7, 2, 7, 5, 4,13, 7, 7, 10, 10, 0, 5, 3, 9 and 8. For this data set, find :

a) Mean and standard deviation, median and rangeb) Display the data in the histogram plot, dot diagram and also stem-and-leaf

diagramc) Determine the intervals x ± s, x ± 2s, x ± 3sd) Find the proportion of the meausurements that lie in each of this intervals.e) Compare your finding with empirical guideline of bell-shaped distribution

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3. The data below were obtained from the detailed record of purchases over several month. The usage vegetables (in weeks) for a household taken from consumer panel were (gram) :

84 58 62 65 75 76 56 87 68 77 87 55 65 66 76 78 74 81 83 78 75 74 60 50 86 80 81 78 74 87

a. Plot a histogram of the data! b. Find the relative frequency of the usage time that did not exceed

80. c. Calculate the mean, variance and the standard deviation d. Calculate the median and quartiles.

4. The mean of corn weight is 278 g by ear and deviation standard is 9,64 g, and than we have 10 ears. If they are gotten from ten different fields, mean of plant height is Rp. 1200,- and its deviation standard is Rp 90,-, which one have more homogenous, the weight of corn ear or the plant height? Explain your answer! Verify your results by direct calculation with the other data.

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5. The employment’s salary at seed company, abbreviated, as follows : 18, 15, 21, 19, 13, 15, 14, 23, 18 and 16 rupiah. If these abbreviation is real salary divide Rp. 100.000,-, find the mean, variance and deviation standard of them.

 6. Computer-aided statistical calculations. Calculation of the

descriptive statistic such as x and s are increasingly tedious with large data sets. Modern computers have come a long way in alleviating the drudgery of hand calculation. Microsoft Exel, Minitab or SPSS are three of computing packages those are easy accessible to student because its commands are in simple English. Find these programs and install its at your computers. Bellow main and sub menu of Microsoft Exel, Minitab and SPSS program. Use these software to find x, s, s2, and coefisien of variation (CV) for data set in exercise b. Histogram and another illustration can also be created.

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7. Some properties of the standard deviationa) if a fixed number c is added to all measurements in a data

set, will the deviations (xi -x) remain changed? And consequently, will s² and s remain changed, too? Take data sample.

b) If all measurements in a data set are multiplied by a fixed number d, the deviation (xi -x) get multiplied by d. Is it right? What about the s² and s? Take data sample.

c) Apply your computer software to explain your data sample. Verify your results by other data.

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