[Chapter 5] z-Scores: Location of Scores and Standardized Distribution

   5.1 Introduction to z-Scores

• Raw Scores : original, unchanged scores that are direct result of mesurement
• a score by itself does not necessarily provide much information about its position within a distribution
• Z-Scores tell exactly where the original scores are located.

// We transform X values into z-scores.. z-Score makes raw scores more meaningful..

   Interpreting a Raw Score

	English	Mathematics	Psychology
X	80	65	75

▲ Raw score

	English	Mathematics	Psychology
X	80	65	75
μ (mean)	85	55	60
σ (SD)	10	5	15

• Raw scores cannot be compared directly because they come from distributions with different means and different standard deviations
• // 날 것의 점수인 과목별 점수를 똑같이 비교할 수 없음. 다른 평균과 다른 표준편차를 가진 분포를 가졌기 때문이다.
• Transforming the scores on each test to a common scale with a specified mean and standard deviation, we can compare the scores
• //각각의 과목시험의 점수들을 지정된 평균과 표준 편차가 있는 공통 척도로 변환하여 점수를 비교할 수 있습니다.

     Standard Scores (Z scores)

Z-score: convert the original scores to new scores with a mean of 0 and a standard deviation of 1 for easy comparison among different scores from different distributions.

mean = 0, SD = 1

 

  Definition:

The sign of the z-score (+ or -) signifies whether the score is above the mean (positive) or below the mean (negative). The numerical value of the z-score specifies the distance from the mean by counting the number of standard deviations between X and  μ .

z-점수의 부호(+ 또는 -)는 점수가 평균보다 높은지(양수) 평균보다 낮은지(음수)를 나타냅니다. 

z-점수의 숫자값은 X 와 μ 사이의 표준편차 수을 계산하여 평균으로부터의 거리를 지정합니다.

 

   Advantage of z-score

A z-score specifies the precise location of each X value within a distribution.

• the mean is zero
  • Positive/negative z scores => above/below the average
• the standard deviation is 1
• Shape of the distribution do not change

//위에는 PPT 내용..

   Z-score formula

   Z-scores

	English	Mathematics	Psychology
X	80	65	75
μ (mean)	85	55	60
σ (SD)	10	5	15
z	-0.5	+2	+1

• 'z-score= +2' means : the score is located above the mean by exactly 2 standard deviations.
• 'z-score= -0.5' means : the score is located below the mean by exactly 1/2 standard deviations.

    Z-scores Example1

In a certain city the mean price of a quart of milk is 63 cents and the standard deviation is 8 cents. The average price of a package of bacon is $1.80 and the standard deviation is 15 cents. If we pay $0.89 for a quart of milk and $2.19 for a package of bacon at a 24-hour convenience store, which is relatively more expensive?

 

a quart of milk

mean =0.63    SD=0.08

 

a package of bacon 

mean=1.80     SD=0.15

 

0.89 for milk

2.19 for bacon

 

z-score for milk = 0.26/0.08= 3.25

z-score for bacon = 0.39/0.15=2.6

 

milk is relatively more expensive

 

   Disadvantage of z-score

It is difficult to explain to someone who is not well versed in statistics

통계에 정통하지 않은 사람에게 설명하기가 어렵다.

  Ex Z score of 0 doesn’t mean that you received a test score of zero, but your score is equal to the average of the class 

Z 점수가 0이라는 것은 시험 점수가 0이라는 것을 의미하지는 않지만 점수는 수업 평균과 같다.

 

     T scores

• T scores: a set of scores with a mean of 50 and a standard deviation of 10
• Formula: T = 10Z + 50
• Each raw score is converted to a Z score, each Z score is multiplied by 10 and 50 is added to each resulting score
• Since the mean of T score is 50, easy to tell a score is above average, or below average
• Standard deviation is 10, so easy to tell how many standard deviation s above or below average a score is

	English	Mathematics	Psychology
X	80	65	75
μ (mean)	85	55	60
σ (SD)	10	5	15
z	-0.5	+2	+1
T	45	70	60

     SAT Scores

• SAT & GRE: Standardized Test
• Formula
  • SAT=100Z + 500
• Scores are transformed to a scale with a mean of 500 and a standard deviation of 100
• Ex) SAT score of 642
• 642 = 100Z+500
• Z=1.42

 

 

 

 

▼ 이해에 도움이 될 유용한 설명 자료

https://drhongdatanote.tistory.com/50?category=648822 

[개념 통계 14] 정규분포와 표준정규분포 그리고 Z-score

▼ z-table 과 t-table 보는 법

https://m.blog.naver.com/PostView.naver?isHttpsRedirect=true&blogId=21thjojo&logNo=110181047644 

표준정규분포표 보는법

https://math100.tistory.com/43

t분포표 보는 법