Build spread from deviations, not a memorized keystroke

1. 1
   Find the center

   The sample mean is 3.

   $$\bar{x}=3$$
2. 2
   Sum squared deviations

   The deviations are negative 2, 0, and 2.

   $$(-2)^2+0^2+2^2=8$$
3. 3
   Use n minus 1

   Divide by 2 and take the nonnegative square root.

   $$s=\sqrt{8/2}=2$$

Independent check: The identity (n-1)s² = 2(4) returns the squared-deviation total 8.

1. 1
   Find the population mean

   The total is 16 across four values.

   $$\mu=16/4=4$$
2. 2
   Average squared deviations

   Four squared deviations of 4 total 16.

   $$\sigma^2=16/4=4$$
3. 3
   Take the square root

   Standard deviation is nonnegative.

   $$\sigma=\sqrt4=2$$

Independent check: The shortcut E(X²) - μ² gives 20 - 16 = 4 for the population variance.

1. 1
   Center and deviations

   The mean is 3 and both deviations have magnitude 2.

   $$\sum(x_i-3)^2=4+4=8$$
2. 2
   Population result

   Divide by N = 2.

   $$\sigma=\sqrt{8/2}=2$$
3. 3
   Sample result

   Divide by n - 1 = 1.

   $$s=\sqrt8=2\sqrt2$$

Independent check: Squaring each answer and multiplying by its denominator returns 8.

1. 1
   Transform the mean

   The mean receives the same shift.

   $$\mu_Y=\mu_X+12$$
2. 2
   Compare deviations

   The added constants cancel.

   $$Y-\mu_Y=(X+12)-(\mu_X+12)=X-\mu_X$$

Independent check: The range also stays unchanged because both endpoints increase by 12.

1. 1
   Ignore the shift for spread

   Adding 10 moves the center only.

   $$\sigma_{a+bX}=|b|\sigma_X$$
2. 2
   Use the absolute scale

   Standard deviation cannot inherit the negative sign.

   $$\sigma_Y=|-3|(4)=12$$

Independent check: Two X-values one SD apart differ by 4; their transformed Y-values differ by 12.

1. 1
   Use the zero condition

   A sum of squared deviations is zero only if each deviation is zero.

   $$\sum(x_i-\bar{x})^2=0$$
2. 2
   Inspect the sets

   Only A contains one repeated value and no variation.

   $$4-4=0$$

Independent check: Sets B and C each contain at least two distinct values, so each has a positive squared deviation.

1. 1
   Use the defining relationship

   Sample standard deviation is the square root of sample variance.

   $$s=\sqrt{s^2}$$
2. 2
   Square

   Restore the variance units.

   $$s^2=(2.5)^2=6.25$$

Independent check: The nonnegative square root of 6.25 is 2.5.

1. 1
   Square the SD

   The sample variance is 9.

   $$s^2=9$$
2. 2
   Restore the numerator

   There are five sample degrees of freedom.

   $$\sum(x_i-\bar{x})^2=(6-1)(9)=45$$

Independent check: Putting 45 back over 5 gives variance 9 and SD 3.

1. 1
   Compare distances

   The original values lie within two units; 80 is far from the new center.

   $$(80-\bar{x}_{\rm new})^2\gg(10-\bar{x}_{\rm old})^2$$
2. 2
   Use the squared-deviation definition

   The large new squared term raises the average squared distance.

   $$\operatorname{SD}_{\rm new}>\operatorname{SD}_{\rm old}$$

Independent check: The range jumps from 2 to 72, independently confirming a major increase in overall spread.

1. 1
   Find both means

   Average the values and their squares separately.

   $$\mu=\frac{14}{3},\quad E(X^2)=\frac{4+16+64}{3}=28$$
2. 2
   Apply the identity

   Subtract the squared mean.

   $$\sigma^2=28-\left(\frac{14}{3}\right)^2=\frac{56}{9}$$

Independent check: Direct deviations from 14/3 have squared total 56/3; dividing by 3 also gives 56/9.