[Chapter 4] Variability 변산성

  4.1  Introduction to Variability

Definition : Variability provides a quantitative measure of the differences between scroes in a distribution and describes the degree to which the scores are spread out or clustered together.

변산성은 분포에서 점수 간의 차이에 대한 정량적 측정을 제공하고 점수가 분산되거나 함께 클러스터링되는 정도를 설명합니다.

변산성(變散性,variability)은 한 분포에 위치하는 여러 점수들이 집중 경향에서 퍼져 있는 성질을 전제하면 '일정한 범위(range)에 흩어져 퍼져 있는 값'으로 정의할 수 있다.
출처 : 위키백과

• How spread out or scattered the scores in a distribution are
  • Ex 1) Test score of average of 80
    • High variability vs. Low variability
      

      

  • Ex 2) Ambulance response time

    Variability 의의

• Just knowing the “midpoint” or “average” does not tell the entire story
• Two very different DATA ARRAYS could have the same average, but look very different:
  1. 4, 5, 4, 4, 5, 5, 4, 5, 4, 5 M=4.5
  2. 1, 7, 1, 1, 7, 7, 1, 10, 5, 5 M=4.5

      The Range  폭

Definition: the distance covered by the scores ina distribution, from the smalles score to the largest score.

• R= Xmax - Xmin //The symbol R is used for the ragne 
  the largest score minus the smallest score
• EX) 92, 98, 64, 47, 89, 76, 54, 68, 71, 50
  Range? 98 - 47 = 51
• a distribution with an extreme number: poor measure of the variability
  극단적인 숫자가 있는 분포 : 변산성의 취약한 측정 //이 번역.. 괜찮은겨?

  4.2  Defining Standard Deviation and Variance

Deviation is distance from the mean:

deviation score = X -(mean)

편차 = 측정값 - 평균

 

Variance equals the mean of the sqared deviations.

Variance is the average sqared distance from the mean.

분산 = 편차 제곱의 평균

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      Variance 분산

Backgournd:

We also need to know how much to the data points vary from the mean

• The VARIANCE of a group of scores tells you how spread out the scores are around the mean
• Variance is a numerical description of spread
• It uses "deviance scores" - how much a score differs from the mean
• Deviation of a single score is its distance from the mean

    Variance : Population   분산 : 모집단

•  

  

  강의 PPT

  

    Variance : Sample   분산 : 표본

the data of a sample are used to estimate the variance of the population from which the sample was drawn, the population variance estimate is computed instead

표본 데이터는 표본이 추출된 모집단의 분산을 추정하는 데 사용되며, 대신 모집단 분산의 추정치가 계산됩니다.

 

   Standard Deviation  표준 편차

Standard Deviation = Root of Variance 분산의 제곱근

• Most common way of describing the spread of a group of scores
• Steps for computing the standard deviation:
  1. Figure the variance
  2. Take the square root

모집단 분산과 표본 분산은 약간 차이 있음 (unbiased statistic)

 

Example 1

Consider the following data to constitute the population: 10, 60, 50, 30, 40, 20

Find the mean, variance, and SD (Standard Deviation)

 

mean=35, variance=291.6... 

SD=17.07825128... 

 

Describe the Following Raw Data

10 People’s scores on an IQ test:

100 125 95 110 82 100 105 90 84 109

Calculate mean, median, mode, range, variance, SD

 

82

84

90

95

100

100

105

109

110

125

 

N=10

mean = 100

median = (100+100) /2 = 100

mode = 100

range = 125-82 = 43

variance = 153.6

SD = 12.39354671...

 

 

  4.5  Sample Variance as an Unbiased Statistic

Sample variability tends to underestimate the variability in the corresponding population. 

To correct for this problem we adjusted the formula for sample variance by dividing by (N - 1) instead of dividing by N. 

The result of the adjustment is that sample variance provides a much more accurate representation of the population variance. (후략)

 

표본 분산은 모집단의 분산을 작게 측정하는 경향이 있다.

그래서 마련된 해결책이, 폭 N 대신 N-1을 나누는 것이다. 이러한 조정이 표본의 분산이 모집단의 분산에 가깝게 더욱 정확한 예측이 가능하게 한다.

사실 어떤 것은 과소측정을 할 수도 있고 과대측정을 할 수도 있는 것.

하지만, 표본 분산의 평균은 모집단 분산에 가까운 정확한 추정을 할 수 있을 것이다.

이러한 것은 unbiased statistic의 개념에서 나온 것이다.

    Variance : Sample   분산 : 표본

• N – 1 : Unbiased estimate of the population variance
• Conservative: an estimate that on the average will be too large
•  
• However, it is difficult to compare deviation with the distribution since deviation is squared