11.3
선형 상관 계수의 계산 및 해석
1900년대 초에 칼 피어슨이 개발한 상관 계수 r은 수치적이며 독립 변수 x와 종속 변수 y 사이의 선형 연관성의 강도와 방향을 측정합니다. 따라서 피어슨 곱적률 상관 계수라고도 합니다. 다음 방정식을 사용하여 계산할 수 있습니다.
여기서 n = 자료점의 수입니다.
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특정 기간 동안의 연간 온도와 비교한 이산화탄소 수치의 데이터 세트를 생각해 보십시오. 데이터 점의 산점도는 두 변수 간의 가능한 선형 패턴을 보여줍니다.
직선 패턴을 확인하기 위해 선형 상관 계수 r이 계산됩니다.
먼저 x 제곱, y 제곱, x와 y의 곱을 결정한 다음 더합니다. 데이터 점의 수는 7입니다.
이 값에서 상관 계수가 계산됩니다.
상관 계수 값의 의미는 임계 값 테이블을 사용하여 해석할 수 있습니다.
0.05의 유의 수준에서 n이 7과 같으면 임계값은 0.754가 됩니다.
r의 계수가 임계값보다 크기 때문에 변수 사이에 선형 상관 관계가 있다는 결론을 뒷받침하는 충분한 증거가 있습니다.
r 제곱 값은 연간 온도 변화의 76.2%가 이산화탄소 수준과 연간 온도 사이의 선형 관계로 설명될 수 있음을 나타냅니다.
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Q1: What does the linear correlation coefficient measure?
The linear correlation coefficient, r, is a numerical measure developed by Karl Pearson that quantifies the strength and direction of the linear association between two variables. It indicates whether variables move together in a predictable straight-line pattern. Values range from -1 to +1, where values closer to -1 or +1 indicate stronger linear relationships, while values near 0 suggest weak or no linear association.
Q2: How do you determine if a correlation coefficient is statistically significant?
Compare the absolute value of the calculated correlation coefficient to the critical value from a significance table at your chosen significance level. If the absolute value of r exceeds the critical value, the correlation is statistically significant. For example, at a 0.05 significance level with 7 data points, a critical value of 0.754 means any |r| greater than 0.754 indicates significant linear correlation between variables.
Q3: What is the coefficient of determination and what does it tell you?
The coefficient of determination, r², is the square of the correlation coefficient expressed as a percentage. It represents the percent of variation in the dependent variable that can be explained by the independent variable using the regression line. For instance, an r² of 76.2% means 76.2% of temperature variation is explained by carbon dioxide levels, while 23.8% remains unexplained.
Q4: What calculations are needed to compute the linear correlation coefficient?
To calculate r, you must first determine x², y², and the product of x and y for each data point, then sum these values. The correlation coefficient formula uses these sums along with the number of data points, n. These intermediate calculations provide the necessary components to quantify the linear relationship strength between your two variables.
Q5: Why is the linear correlation coefficient also called the Pearson product-moment correlation coefficient?
The coefficient is named after Karl Pearson, who developed it in the early 1900s. The term 'product-moment' refers to the calculation method, which involves computing products of paired values and their deviations from the mean. This naming convention distinguishes it from other correlation measures and honors its mathematical foundation.
Q6: What does it mean when the correlation coefficient falls between the critical values?
When r falls between the positive and negative critical values from the significance table, the correlation coefficient is not statistically significant. This means there is insufficient evidence to conclude a true linear relationship exists between the variables. In such cases, using the regression line for prediction is not recommended.
Q7: How does the unexplained variation relate to data scatter around the regression line?
The unexplained variation, calculated as 1 – r² and expressed as a percentage, represents the portion of dependent variable variation not explained by the regression line. This unexplained variation appears as the scattering of observed data points about the best-fit line, which can be examined using residual plots. Greater scatter indicates more unexplained variation and a weaker linear relationship.