Measures of Central Tendency

A statistic is a characteristic or measure obtained by using the data values from a sample.

statistika adalah karakteristik atau ukuran yang diperoleh dengan menggunakan nilai data dari sampel.

A parameter is a characteristic or measure obtained by using the data values from a specific population.

parameter adalah karakteristik atau ukuran yang diperoleh dengan menggunakan nilai data dari populasi tertentu.

The Mean (arithmetic average)

The mean is defined to be the sum of the data values divided by the total number of values.

Mean = Sum of data values / Total number of values

We will compute two means: one for the sample and one for a finite population of values.

Kita akan menghitung dua mean: satu untuk sampel dan satu untuk populasi nilai yang terbatas.

The mean, in most cases, is not an actual data value.

Mean, dalam kebanyakan kasus, bukan merupakan nilai data yang sebenarnya.

Sample Mean

The symbol

\overline{\rm X}

represents the sample mean.

\overline{\rm X}

is read as “X - bar”.
The Greek symbol

\Sigma

is read as “sigma” and it means “to sum”

\overline{\rm X} = \dfrac{X_1 + X_2 + ... + X_n}{n}

~~~~ = \dfrac{\Sigma X}{n}

Sample Mean - Example

The ages in weeks of a random sample of six kittens at an animal shelter are 3, 8, 5, 12, 14, and 12. Find the average age of this sample.
The sample mean is

\overline{\rm X} = \dfrac{\Sigma X}{n} = \dfrac{3+8+5+12+14+12}{6}

~~~~~~~~~~~~~~~~~ = \dfrac{54}{6} = 9

weeks.

The Population Mean

The Greek symbol

\mu

represents the population mean. The symbol

\mu

is read as “mu”.

N

is the size of the finite population.

\mu = \dfrac{X_1 + X_2 + ... + X_N}{N}

~~~~ = \dfrac{\Sigma X}{N}

Population Mean - Example

A small company consists of the owner the manager, the salesperson and two technicians. The salaries are listed as $50,000, 20,000, 12,000, 9,000 and 9,000 respectively (Assume this is the population)
Then the population mean will be.

\mu = \dfrac{\Sigma X}{N}

~~~~ = \dfrac{50,000 + 20,000 + 12,000 + 9,000 + 9,000}{5} = \$20,000

The Sample Mean for an Ungrouped Frequency Distribution

The mean for an ungrouped frequency distributuion is given by

\overline{\rm X} = \dfrac{\Sigma (f \bullet X)}{n}

Here

f

is the frequency for the corresponding value of

X

and

n = \Sigma f

Sample Mean for an Ungrouped Frequency Distribution - Example

The scores for students on a 4 - point quiz are given in the table.
Find the mean score.

Score ( $X$ )	Frequency ( $f$ )	$f \bullet X$
0	2	0
1	4	4
2	12	24
3	4	12
4	3	12

\overline{\rm X} = \dfrac{\Sigma (f \bullet X)}{n} = \dfrac{52}{25} = 2.08

The Sample Mean for a Grouped Frequency Distribution

The mean for a grouped frequency distributuion is given by

\overline{\rm X} = \dfrac{\Sigma (f \bullet X_m)}{n}

Here

X_m

is the corresponding class midpoint.

Sample Mean for a Grouped Frequency Distribution - Example

Given the table below, find the mean.

Class	Frequency ( $f$ )	Class Midpoint ( $X_m$ )	$f \bullet X_m$
15.5 - 20.5	3	18	54
20.5 - 25.5	5	23	115
25.5 - 30.5	4	28	112
30.5 - 35.5	3	33	99
35.5 - 40.5	2	38	76

\Sigma (f \bullet X_m) = 54 + 115 + 112 + 99 + 76 = 456

and

n = 17.

\overline{\rm X} = \dfrac{\Sigma (f \bullet X_m)}{n} = \dfrac{456}{17} = 26.82

The Median

When a data set is ordered, it is called a data array.

Ketika kumpulan data diurutkan, disebut sebagai data array.

The median is defined to be the midpoint of the data array

median didefinisikan sebagai titik tengah dari data array

The symbol used to denote the median is MD.

Simbol yang digunakan untuk menunjukkan median adalah MD.

The Median - Example odd number

The weights (in pounds) of seven army recruits are 180, 201, 220, 191, 219, 209, and 186. Find the median.

Arrange the data in order and compute the middle point:
Data array: 180, 186, 191, 201, 209, 219, 220.
The median, MD = 201.

The Median - Example even number

When there is an even number of values in the data set, the median is obtained by taking the average of the two middle numbers.

Ketika ada jumlah nilai genap dalam kumpulan data, median diperoleh dengan mengambil rata-rata dari dua angka tengah.

Six customers purchased the following number of magazines: 1, 7, 3, 2, 3, 4. Find the median.

Arrange the data in order and compute the middle point:
Data array: 1, 2, 3, 3, 4, 7.
The median, MD = (3 + 3)/2 = 3.

The ages of 10 college students are: 18, 24, 20, 35, 19, 23, 26, 23, 19, 20. Find the median.

Arrange the data in order and compute the middle point:
Data array: 18, 19, 19, 20, 20, 23, 23, 24, 26, 35.
The median, MD = (20 + 23)/2 = 21.5.

The Median for an Ungrouped Frequency Distribution

For an ungrouped frequency distribution, find the median by examining the cumulative frequencies to locate the middle value.

Untuk distribusi frekuensi yang tidak dikelompokkan, temukan median dengan memeriksa frekuensi kumulatif untuk menemukan nilai tengah.

If n is the sample size, compute n/2. Locate the data point where n/2 values fall below and n/2 values fall above.

Jika n adalah ukuran sampel, hitung n/2. Temukan titik data di mana n/2 nilai berada di bawah dan n/2 nilai berada di atas.

The Median for an Ungrouped Frequency Distribution - Example

LRJ Appliance recorded the number of VCRs sold per week over a one-year period. The data is given below. Find the median.
No. Sets Sold Frequency Cumulative Frequency
1 4 4
2 9 13 $\uparrow$
3 6 19
4 2 21
5 3 24

No. Sets Sold	Frequency	Cumulative Frequency
1	4	4
`2`	9	`13` $\uparrow$
3	6	19
4	2	21
5	3	24

To locate the middle point, divide n by 2: 24/2 = 12.
Locate the point where 12 values would fall below and 12 values will fall above.
Consider the cumulative distribution.
The $12^{th}$ and $13^{th}$ values fall in class 2.
The median, MD = 2.

Median for an Ungrouped Frequency Distribution

The Median for a Grouped Frequency Distribution

The median can be computed from:

MD = \dfrac{n / 2 - cf}{f} (w) + L_m

Where:

$~~~ n~~~=$ sum of the frequencies
$~~~ cf~=$ cumulative frequency of the class before the median class
$~~~ f~~~=$ frequency of the median class
$~~~ w~~=$ width of the median class
$~~~ L_m=$ lower boundary of the median class

The Median for a Grouped Frequency Distribution - Example

Given the table below, find the median.

Class	Frequency ( $f$ )	Cumulative Frequency ( $cf$ )
15.5 - 20.5	3	3
20.5 - 25.5	5	8
`25.5 - 30.5`	4	12
30.5 - 35.5	3	15
35.5 - 40.5	2	17

To locate the halfway point, divide n by 2: 17/2 = 8.5 $\uparrow$ 9.
Find the class that contains the $9^{th}$ value. This will be the median class.
Consider the cumulative distribution.
The median class will then be 25.5 - 30.5.

$n = 17$
$cf = 8$
$f = 4$
$w = 25.5 - 20.5 = 5$
$L_m = 25.5$

MD = \dfrac{n / 2 - cf}{f} (w) + L_m = \dfrac{(17 / 2) - 8}{4} (5) + 25.5

~~~~~~~~~ = 26.125

The Mode (Modus)

The mode is defined to be the value that occurs most often in a data set.

Modus didefinisikan sebagai nilai yang paling sering muncul dalam kumpulan data.

A data set can have more than one mode.

Sebuah kumpulan data dapat memiliki lebih dari satu modus.

A data set is said to have no mode if all values occur with equal frequency.

Sebuah kumpulan data dapat dikatakan tidak memiliki modus jika semua nilai muncul dengan frekuensi yang sama.

The Mode - Example Exist Mode

The following data represent the duration (in days) of U.S. space shuttle voyages for the years 1992-94. Find the mode.
Data set: 8, 9, 9, 14, 8, 8, 10, 7, 6, 9, 7, 8, 10, 14, 11, 8, 14, 11. Ordered set: 6, 7, 7, 8, 8, 8, 8, 8, 9, 9, 9, 10, 10, 11, 11, 14, 14, 14. (Mode = 8).

The Mode - Example No Mode

Six strains of bacteria were tested to see how long they could remain alive outside their normal environment. The time, in minutes, is given below. Find the mode.
Data set: 2, 3, 5, 7, 8, 10. There is no mode since each data value occurs equally with a frequency of one.

The Mode - Example Double/Two Mode

Eleven different automobiles were tested at a speed of 15 mph for stopping distances. The distance, in feet, is given below. Find the mode.
Data set: 15, 18, 18, 18, 20, 22, 24, 24, 24, 26, 26.
There are two modes (bimodal). The values are 18 and 24.
Why?

The Mode for an Ungrouped Frequency Distribution - Example

Given the table below, find themode.

	Values	Frequency ( $f$ )
	15	3
	20	5
Mode $\longmapsto$	`25`	`8`
	30	3
	35	2

The Mode for a Grouped Frequency Distribution - Example

The mode for grouped data is the modal class.

Modus untuk data yang dikelompokkan adalah kelas modal.

The modal class is the class with the largest frequency.

Kelas modal adalah kelas dengan frekuensi terbesar.

Sometimes the midpoint of the class is used rather than the boundaries.

Kadang-kadang titik tengah kelas digunakan daripada batasnya.

Given the table below, find themode.

	Class	Frequency ( $f$ )
	15.5 - 20.5	3
	15.5 - 20.5	5
Modal Class $\longmapsto$	`15.5 - 20.5`	`7`
	30.5 - 35.5	3
	30.5 - 35.5	2

The Midrange

The midrange is found by adding the lowest and highest values in the data set and dividing by 2.

Midrange ditemukan dengan menambahkan nilai terendah dan tertinggi dalam kumpulan data dan membaginya dengan 2.

The midrange is a rough estimate of the middle value of the data.

Midrange adalah perkiraan kasar dari nilai tengah data.

The symbol that is used to represent the midrange is MR.

Simbol yang digunakan untuk mewakili midrange adalah MR.

The Midrange - Example

Last winter, the city of Brownsville, Minnesota, reported the following number of water-line breaks per month.The data is as follows: 2, 3, 6, 8, 4, 1.Find the midrange : MR = (1 + 8)/2 = 4.5.

Note: Extreme values influence the midrange and thus may not be a typical description of the middle.

The Weighted Mean

The weighted mean is used when the values in a data set are not all equally represented.

Mean tertimbang digunakan ketika nilai dalam kumpulan data tidak semuanya diwakili dengan sama.

The weighted mean of a variable X is found by multiplying each value by its corresponding weight and dividing the sum of the products by the sum of the weights.

Mean tertimbang dari variabel X ditemukan dengan mengalikan setiap nilai dengan bobotnya yang sesuai dan membagi jumlah produk dengan jumlah bobot.

The weighted mean

\overline{\rm X} = \dfrac{w_1 X_1 + w_2 X_2 + ... + w_n X_n}{w_1 + w_2 + ... + w_n} = \dfrac{\Sigma (w X)}{\Sigma w}

where

w_1, w_2, ..., w_n

are the weights and

X_1, X_2, ..., X_n

are the values.

Distribution Shapes

Frequency distributions can assume many shapes.

Distribusi frekuensi dapat mengasumsikan banyak bentuk.

Persiapan Belajar

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Measures of Central Tendency