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The mean and the standard deviation: a recap

Statistics, grade 11 (4 units) Lesson 1 of 20Learning

The mean and the standard deviation: a recap

The mean

The mean xˉ\displaystyle \bar{x} (read “x bar”) is the sum of the values divided by the number of values, N\displaystyle N.

Five friends counted how many messages they sent in the last hour: 3, 5, 6, 8, 13.

xˉ=3+5+6+8+135=355=7\bar{x}=\frac{3+5+6+8+13}{5}=\frac{35}{5}=7
The standard deviation

The standard deviation S\displaystyle S measures spread: roughly how far the values are from the mean. It is worked out like this:

  • for each value – its difference from the mean, squared;
  • the mean of the squares;
  • the square root of that mean.

For the friends’ messages the differences from 7 are −4,−2,−1,1,6\displaystyle -4, -2, -1, 1, 6:

S=16+4+1+1+365=11.6≈3.41S=\sqrt{\frac{16+4+1+1+36}{5}}=\sqrt{11.6}\approx3.41
  • A small standard deviation – the values are packed close to the mean; a large one – they are spread out.
  • The standard deviation is in the same units as the values: here, messages.
  • It is 0 only when all the values are equal.
The mean and the standard deviation from a frequency table

When a value appears several times, it – or the square of its difference – is multiplied by its frequency f\displaystyle f, the number of times it appears.

The mark x\displaystyle x60708090100
Number of pupils f\displaystyle f25841

The class has N=20\displaystyle {N}={20} pupils.

xˉ=x1f1+x2f2+⋯+xnfnN\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_nf_n}{N}
S=(x1−xˉ)2f1+(x2−xˉ)2f2+⋯+(xn−xˉ)2fnN\displaystyle S=\sqrt{\frac{(x_1-\bar{x})^{2}f_1+(x_2-\bar{x})^{2}f_2+\dots+(x_n-\bar{x})^{2}f_n}{N}}
  • xˉ=60⋅2+70⋅5+80⋅8+90⋅4+100⋅120=157020=78.5\displaystyle {\bar{x}}={\frac{60\cdot2+70\cdot5+80\cdot8+90\cdot4+100\cdot1}{20}}={\frac{1570}{20}}={78.5}.
  • S=(−18.5)2⋅2+(−8.5)2⋅5+1.52⋅8+11.52⋅4+21.52⋅120=205520≈10.14\displaystyle {S}={\sqrt{\frac{(-18.5)^{2}\cdot2+(-8.5)^{2}\cdot5+1.5^{2}\cdot8+11.5^{2}\cdot4+21.5^{2}\cdot1}{20}}}={\sqrt{\frac{2055}{20}}\approx10.14}.
Changing every value

The teacher added 5 points to every mark in this class.

  • Every mark moved up by 5, so the mean moved up by 5 too: 78.5+5=83.5\displaystyle {78.5+5}={83.5}.
  • The distances between the marks and the mean did not change, so the standard deviation stays 10.14\displaystyle 10.14.

Had the teacher multiplied every mark by 1.1, every distance from the mean would have grown 1.1 times:

  • the mean: 78.5⋅1.1=86.35\displaystyle {78.5\cdot1.1}={86.35};
  • the standard deviation: 10.14⋅1.1≈11.15\displaystyle 10.14\cdot1.1\approx11.15.

In short: adding a number to every value moves the mean and leaves the standard deviation as it is; multiplying every value by a positive number multiplies both by that number.

Exercise 1

The times, in minutes, five pupils took to get to school: 4, 7, 7, 9, 13.

a

Work out the mean.

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

    Work out the standard deviation, to two decimal places.

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      Exercise 2

      20 pupils were asked how many books they read last month.

      Number of books x\displaystyle x01234
      Number of pupils f\displaystyle f36731
      a

      Work out the mean.

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        b

        Work out the standard deviation, to two decimal places.

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 3

          In a factory the mean monthly pay is 8,000 ₪, and the standard deviation 1,200 ₪.

          a

          Every worker got a rise of 200 ₪. What are the new mean and the new standard deviation? Explain.

          xˉ\displaystyle \bar{x}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            S\displaystyle S

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              b

              Instead of that rise, every worker got a 10% rise. Work out the new mean.

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                c

                Work out the standard deviation after the 10% rise.

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power