प्लान

रैखिक समाश्रयण कैलकुलेटर (सर्वोत्तम आसंजन रेखा)

यह कैलकुलेटर युग्मित आँकड़ों के लिए न्यूनतम वर्ग वाली सर्वोत्तम आसंजन रेखा y = mx + b निकालता है। यह माध्य, प्रयुक्त योग, ढलान और अंतःखंड, तथा सहसंबंध गुणांक r और r² दिखाता है, जो बताते हैं कि रेखा कितनी अच्छी तरह बैठती है।

मानों को अल्पविराम या स्पेस से अलग करें।

नतीजा

मान डालें और नतीजा देखने के लिए “गणना करें” दबाएँ।

सूत्र

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.

यह कैलकुलेटर कैसे इस्तेमाल करें

  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.

हल किए गए उदाहरण

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.

आम गलतियाँ

  • 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).

अक्सर पूछे जाने वाले सवाल

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.