Strength of antibiotics-bacteria relationship

The strength and direction of a linear relationship are measured by Karl Pearson's correlation coefficient r. Let x = antibiotics (mg), y = bacteria.

Sums (n = 6):

        x        y       xy       x²       y²
       12        5       60      144       25
       15        7      105      225       49
       14      5.6     78.4      196    31.36
       16      7.2    115.2      256    51.84
       17      8.6    146.2      289    73.96
       10      6.2       62      100    38.44
---------------------------------------------
       84     39.6    566.8     1210    269.6

Σx = 84, Σy = 39.6, Σxy = 566.8, Σx² = 1210, Σy² = 269.6

Means: x̄ = 14.0000, ȳ = 6.6000

Sxx = Σx² − (Σx)²/n = 34.0000 Sxy = Σxy − (Σx)(Σy)/n = 12.4000 Syy = Σy² − (Σy)²/n = 8.2400

Slope b = Sxy/Sxx = 12.4000/34.0000 = 0.3647 Intercept a = ȳ − bx̄ = 6.6000 − (0.3647)(14.0000) = 1.4941

r = Sxy/√(Sxx·Syy) = 12.4000/√(34.0000 × 8.2400) = 12.4000/16.7380 = 0.7408

Direction: r is positive, so the relationship is direct: as the antibiotic dose increases, the bacteria measure also increases.

Strength: r = 0.741 indicates a moderately strong to strong positive linear relationship (r² = 0.549, so 54.9% of the variation in bacteria is explained by the antibiotic amount; the rest is due to other factors).