Kuswanto 2007

Ukuran keragamanUkuran keragaman

Dari tiga ukuran pemusatan, belum dapat memberikan Dari tiga ukuran pemusatan, belum dapat memberikan deskripsi yang lengkap bagi suatu data. deskripsi yang lengkap bagi suatu data.

Perlu juga diketahui seberapa jauh pengamatan-Perlu juga diketahui seberapa jauh pengamatan-pengamatan tersebut menyebar dari rata-ratanya. pengamatan tersebut menyebar dari rata-ratanya.

Ada kemungkinan diperoleh rata-rata dan median yang Ada kemungkinan diperoleh rata-rata dan median yang sama, namun berbeda keragamannya.sama, namun berbeda keragamannya.

Beberapa ukuran keragaman yang sering kita temui Beberapa ukuran keragaman yang sering kita temui adalah range (rentang=kisaran=wilayah), simpangan adalah range (rentang=kisaran=wilayah), simpangan (deviasi), varian (ragam), simpangan baku (standar (deviasi), varian (ragam), simpangan baku (standar deviasi) dan koefisien keragaman.deviasi) dan koefisien keragaman.

Measures of Dispersion and VariabilityMeasures of Dispersion and Variability

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

f

X

f

X

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

2. DEVIATION DEVIASI = SIMPANGAN

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

Sample mean deviation = | Xi - X |

nEssentially the average deviation from the mean

3. Mean Deviation = Simpangan Rerata

4. Variance = Ragam

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/nn - 1

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

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

n - 1 =

5. 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

Example

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

= 1.304

4

6. 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%

Example:

22345

= X1

= X2

= X3

= X4

= X5

s = 1.304 g

CV =sX

X = 3.2 g

CV = 1.304 g

3.2 g

CV = 0.4075

or

CV = 40.75%

(X 100%)

Attention there is not any UNIT, or %

7. Probability (Peluang) :

Likelihood of an event - represented by Pnumber between 0 and 1

Eg coin toss: probability of heads = 0.5

Eg 2: roll of die: probability of any number = 1/6

So probability of one event = 1

# of possible outcomes

Adding and multiplying probabilities:

To find prob. Of one event and another event both happening, multiply the probabilities of the two events

(0.5)(0.5) = 0.25

eg if a coin is tossed twice, what is prob of a head followed by a tail

To find prob. of one event or another event happening, add the probabilities of the two events

eg if a die is tossed what is prob of rolling a 2 or a 4?

1/6 + 1/6= 2/6= .333

BUT : The biological world is not as clear cut as tossing coins

Relative frequency of an event = frequency of that eventtotal # of all events

We can guess at a probability by sampling a large data set and expressing relative frequency

Eg. Sample 1000 students, get 510 male510/1000 = 51% male

Relative Frequency Example 2

Vertebrate Number Rel. FreqAmphibians 53 0.06Turtles 41 0.05Snakes 204 0.24Birds 418 0.49Mammals 136 0.16

Total 852 1.00

53/852 = 0.06

Probability that next animal will be a snakeP = 0.24

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 2 323

68.27%

95.44%

99.73%

f

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

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

9. Symmetry and Kurtosis

Symmetry 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

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

Symmetry

ƒ

mode

median

mean

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

Soal dikerjakanSoal dikerjakan

1.1. Banyaknya gol yang dibuat tim Banyaknya gol yang dibuat tim Singo EdanSingo Edan pada musim kompetisi tahun lalu adalah 4, 9, pada musim kompetisi tahun lalu adalah 4, 9, 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, dan 0, 1, 3, 24, 12, 3, 30, 12, 7, 13, 18, 4, 5, dan 15. 15. Dengan menganggap data tersebut Dengan menganggap data tersebut sebagai contoh, hitunglah varian, simpangan sebagai contoh, hitunglah varian, simpangan baku dan koefisien keragamannyabaku dan koefisien keragamannya..

2.2. The mean of snacks weight is 278 g by pack The mean of snacks weight is 278 g by pack and deviation standard is 9,64 g, and than we and deviation standard is 9,64 g, and than we have 10 packs. If they are bought from ten have 10 packs. If they are bought from ten different stores, mean of price is Rp. 1200,- different stores, mean of price is Rp. 1200,- and its deviation standard is Rp 90,-, which and its deviation standard is Rp 90,-, which one have more homogenous, the weight or the one have more homogenous, the weight or the price. price. Explain your answer.Explain your answer.

Soal dikerjakanSoal dikerjakan

3.3. Some properties of the standard deviationSome properties of the standard deviation If a fixed number c is added to all measurements in a If a fixed number c is added to all measurements in a

data set, will the deviations remain changed? And data set, will the deviations remain changed? And consequentyl, will s² and s remain changed, too?consequentyl, will s² and s remain changed, too?

If all measurements in a data set are multiplied by a If all measurements in a data set are multiplied by a fixed number d, the deviation get multiplied by d. Is it fixed number d, the deviation get multiplied by d. Is it right? What about the s² and s? right? What about the s² and s?

4.4. The teacher’s salary, abbreviated, as follows : 18, The teacher’s salary, abbreviated, as follows : 18, 15, 21, 19, 13, 15, 14, 23, 18 and 16 rupiah. If 15, 21, 19, 13, 15, 14, 23, 18 and 16 rupiah. If these abbreviation is real salary divide Rp. these abbreviation is real salary divide Rp. 100.000,-, find the variance of them.100.000,-, find the variance of them.