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The total of all observation or data point values divided by the number of observations.

Formula to calculate the sample mean of the data set of distinct numbers,

$$\begin{gathered} \stackrel{¯}{x}=\frac{\underset{}{∑}x}{n} \\ \stackrel{¯}{x}=\frac{3+8+(-1)+2+0+5+(-3)+1+(-1)+6+5+(-2)}{12} \\ \stackrel{¯}{x}=\frac{30-7}{12}=\frac{23}{12} \\ \stackrel{¯}{x}=1.9167 \end{gathered}$$

The mean change score =$1.92$.

The standard deviation is a measure that indicates how much variation

(such as spread, dispersion, and spread) is there from the mean.

Estimated value for standard deviation of given population with only sample data is sample standard deviation.

Formula :

$$\begin{gathered} S=\sqrt{\frac{\underset{}{∑}{\left(x-\stackrel{¯}{x}\right)}^2}{n-1}} \\ S=\sqrt{\frac{{(3-1.92)}^2+{\left(8-1.92\right)}^2+{\left(-1-1.92\right)}^2+{\left(2-1.92\right)}^2+{\left(0-1.92\right)}^2+{\left(5-1.92\right)}^2+{\left(-3-1.92\right)}^2+{\left(1-1.92\right)}^2+{\left(-1-1.92\right)}^2+{\left(6-1.92\right)}^2+{\left(5-1.92\right)}^2+{\left(-2-1.92\right)}^2}{12-1}} \\ S=\sqrt{\frac{1.{08}^2+6.{08}^2+{(}_{2.92}^{-}+0.8^2+{(}_{1.92}^{-}+3.{08}^2+{(}_{4.92}^{-}+{(}_{0.92}^{-}+{(}_{2.92}^{-}+4.{08}^2+3.{08}^2+{(}_{3.92}^{-}}{11}} \\ S=\sqrt{\frac{1.1664+36.9664+8.5264+0.64+3.6864+9.4864+24.2064+0.8464+8.5264+16.6464+9.4864+15.3664}{11}}=\sqrt{\frac{135.5504}{11}}=\sqrt{12.3328} \\ S=3.5118 \end{gathered}$$

Hence, standard deviation =$3.51$

The middle value in a set of statistics or data is known as the median.

No of observation = n,

Median = M,

Data arranged in ascending order.

$-3,-2,-1,0,1,2,3,5,5,6,8$

Then,

$M=\frac{12+1}{2}th$observation,

$M=7.5th$observation,

M = $7th$observation + $0.5(8th-7th)$observation,

M = $2+(3-2)×0.5=2+0.5$

M = $2.5$

Hence median =$2.5$.

A declaration that two mathematical expressions have the same values.

Equation to calculate the value of the change score which is $2.2$standard deviation down the mean.

value = mean + (#ofSTDVs) (STDV)

value =$\stackrel{¯}{x}$+ (#ofSTDVs)$×$S

Required value be $x_0$,

$$\begin{gathered} x_0=1.92+\left(-2.2\right)×3.51 \\ {x}_0=1.92-7.722 \\ {x}_0=-5.802 \end{gathered}$$

Hence, the value of change score is $-5.8$.