Kuswanto 2012

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.

Measures of Dispersion and Variability

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

f

X

f

X

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

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

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/n

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 %

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

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 range

b) Display the data in the histogram plot, dot diagram and also stem-and-leaf diagram

c) Determine the intervals x ± s, x ± 2s, x ± 3s

d) Find the proportion of the meausurements that lie in each of this intervals.

e) Compare your finding with empirical guideline of bell-shaped distribution

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.

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.

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.