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Formulas: science and society, grade 11 (3 units)

Growth and decay

  • An exponential process – M0\displaystyle M_0 the amount at the start, q\displaystyle q the factor the amount is multiplied by in each unit of time, Mt\displaystyle M_t the amount after t\displaystyle t units of time
    Mt=M0⋅qt\displaystyle M_t=M_0\cdot q^{t}
  • The growth factor – a growth of p\displaystyle p percent in each unit of time
    q=1+p100\displaystyle q=1+\frac{p}{100}
  • The decay factor – a decay of p\displaystyle p percent in each unit of time
    q=1−p100\displaystyle q=1-\frac{p}{100}
  • Decay by half-life – n\displaystyle n the number of half-lives gone by
    Mt=M0⋅(12)n\displaystyle M_t=M_0\cdot\left(\frac{1}{2}\right)^{n}
  • The number of half-lives – t\displaystyle t the time gone by, T\displaystyle T the half-life
    n=tT\displaystyle n=\frac{t}{T}
  • Linear growth – the amount grows by d\displaystyle d in each unit of time
    Mt=M0+d⋅t\displaystyle M_t=M_0+d\cdot t

Measures of spread

  • The mean of n\displaystyle n data
    xˉ=x1+x2+⋯+xnn\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}
  • The range – the greatest datum minus the least
    xmax⁡−xmin⁡\displaystyle x_{\max}-x_{\min}
  • The deviation of the datum xi\displaystyle x_i from the mean
    xi−xˉ\displaystyle x_i-\bar{x}
  • The standard deviation of n\displaystyle n data
    σ=(x1−xˉ)2+(x2−xˉ)2+⋯+(xn−xˉ)2n\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}}
  • The standard deviation from a frequency table – f1,…,fk\displaystyle f_1, \dots, f_k the frequencies of the values x1,…,xk\displaystyle x_1, \dots, x_k
    σ=f1(x1−xˉ)2+f2(x2−xˉ)2+⋯+fk(xk−xˉ)2f1+f2+⋯+fk\displaystyle \sigma=\sqrt{\frac{f_1(x_1-\bar{x})^2+f_2(x_2-\bar{x})^2+\dots+f_k(x_k-\bar{x})^2}{f_1+f_2+\dots+f_k}}

Probability

  • The complement A‾\displaystyle \overline{A} – A\displaystyle A does not happen
    P(A‾)\displaystyle P(\overline{A}) =\displaystyle = 1−P(A)\displaystyle 1-P(A)
  • A path of a tree diagram – the product of the probabilities on its branches
    P=p1⋅p2⋅p3\displaystyle P=p_1\cdot p_2\cdot p_3
  • “At least once” in n\displaystyle n independent experiments – p\displaystyle p the probability that it happens in each
    1−(1−p)n\displaystyle 1-(1-p)^{n}
  • Conditional probability – the probability of A\displaystyle A when B\displaystyle B is known to have happened; A∩B\displaystyle A\cap B – both A\displaystyle A and B\displaystyle B
    P(A∣B)=P(A∩B)P(B)\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}
  • The probability of “both B\displaystyle B and A\displaystyle A”
    P(A∩B)=P(B)⋅P(A∣B)\displaystyle P(A\cap B)=P(B)\cdot P(A|B)
  • Independent events – knowing that B\displaystyle B happened does not change the probability of A\displaystyle A
    P(A∣B)=P(A)\displaystyle P(A|B)=P(A)
  • The expected value – x1,…,xk\displaystyle x_1, \dots, x_k the possible values, p1,…,pk\displaystyle p_1, \dots, p_k their probabilities
    E=x1p1+x2p2+⋯+xkpk\displaystyle E=x_1p_1+x_2p_2+\dots+x_kp_k