요금제

선형 회귀 계산기 (최적 직선)

이 계산기는 짝지은 데이터에 대해 최소제곱법으로 최적 직선 y = mx + b를 구합니다. 평균, 사용한 합, 기울기와 절편, 그리고 직선이 얼마나 잘 맞는지 알려 주는 상관계수 r과 r²을 보여 줍니다.

값은 쉼표나 공백으로 구분하세요.

결과

값을 입력하고 계산을 누르면 결과가 표시됩니다.

공식

m=∑(x−xˉ)(y−yˉ)∑(x−xˉ)2,b=yˉ−mxˉm = \frac{\sum (x - \bar{x})(y - \bar{y})}{\sum (x - \bar{x})^2}, \quad b = \bar{y} - m\bar{x}

The least-squares slope and intercept. The line always passes through the point of means (x̄, ȳ).

r=SxySxxSyyr = \frac{S_{xy}}{\sqrt{S_{xx} S_{yy}}}

r is between −1 and 1: near ±1 means the points lie close to a line; near 0 means no linear pattern.

이 계산기 사용법

  1. Enter the x values and the matching y values in the same order.
  2. Press Calculate to see the equation of the line, the slope, the intercept, r and r².

풀이 예제

x: 1, 2, 3, 4, 5 and y: 2, 4, 5, 4, 5

x̄ = 3 and ȳ = 4. Sxx = 10 and Sxy = 6, so m = 6/10 = 0.6.

b = 4 − 0.6 × 3 = 2.2, so the line is y = 0.6x + 2.2.

Syy = 6, so r = 6/√60 ≈ 0.7746 and r² = 0.6.

A perfect line: y = 2x + 1

Every point lies on y = 2x + 1, so r = 1 and r² = 1.

A downward trend

x̄ = 2.5, ȳ = 7, Sxx = 5 and Sxy = −11, so m = −2.2 and b = 7 + 2.2 × 2.5 = 12.5.

Syy = 26, so r = −11/√130 ≈ −0.9648: a strong downward trend.

자주 하는 실수

  • Entering x and y values in different orders, which pairs the wrong points.
  • Reading r² as the slope: r² is how much of the variation the line explains.
  • Using the line far outside the range of the data (extrapolation).

자주 묻는 질문

How do you calculate the line of best fit?

Find the means of x and y, then m = Sxy/Sxx and b = ȳ − m·x̄. The calculator shows each of these sums.

What does the correlation coefficient r mean?

r measures how close the points are to a straight line: 1 is a perfect upward line, −1 a perfect downward line, 0 no linear relationship.

What is r²?

The share of the variation in y that the line explains. r² = 0.6 means the line explains 60% of it.