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$Given the dara below, compute for the iff. a. bar(x) b. tilde(x) c. hat(x) f. R g. MAD d. theta_(1) h. IQR e. Q_(3) i. variance j. standard deviation {:[23,4,6,7,8999,9,10,13,15,18]:}$

Given the dara below, compute for the iff. $\newline$ a. $\bar{x}$ $\newline$ b. $\tilde{x}$ $\newline$ c. $\hat{x}$ $\newline$ f. $R$ $\newline$ g. $M A D$ $\newline$ d. $\theta_{1}$ $\newline$ h. $I Q R$ $\newline$ e. $Q_{3}$ $\newline$ i. variance $\newline$ j. standard deviation $\newline$ $\begin{array}{llllllllllllll}23 & 4 & 6 & 7 & 8999 & 9 & 10 & 13 & 15 & 18\end{array}$

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Precalculus

Variance and standard deviation

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Q. Given the dara below, compute for the iff.

\newline

\bar{x}

\newline

\tilde{x}

\newline

\hat{x}

\newline

R

\newline

M A D

\newline

\theta_{1}

\newline

I Q R

\newline

Q_{3}

\newline

i. variance

\newline

j. standard deviation

\newline

\begin{array}{llllllllllllll}23 & 4 & 6 & 7 & 8999 & 9 & 10 & 13 & 15 & 18\end{array}

Compute Mean: To compute the mean $\bar{x}$ , we need to sum all the values and divide by the number of values. $\newline$ Sum of all values: $23 + 4 + 6 + 7 + 8999 + 9 + 10 + 13 + 15 + 18 = 9104$ $\newline$ Number of values $n$ : $10$ $\newline$ Mean $\bar{x}$ = Sum of all values / $n$ = $9104 / 10 = 910.4$
Compute Median: To compute the median ( $\tilde{x}$ ), we need to arrange the data in ascending order and find the middle value. If the number of values is even, the median is the average of the two middle numbers. $\newline$ Ordered data: $4, 6, 7, 9, 10, 13, 15, 18, 23, 8999$ $\newline$ Since there are $10$ values, the median will be the average of the $5$ th and $6$ th values. $\newline$ Median ( $\tilde{x}$ ) = $(10 + 13) / 2 = 23 / 2 = 11.5$
Compute Mode: To compute the mode $\hat{x}$ , we need to identify the value that appears most frequently in the data set. If all values appear only once, there is no mode. $\newline$ In this data set, all values appear only once, so there is no mode. $\newline$ Mode $\hat{x}$ = None
Compute Range: To compute the range $R$ , we subtract the smallest value from the largest value. $\newline$ Range $R$ = Largest value - Smallest value = $8999 - 4 = 8995$
Compute MAD: To compute the Mean Absolute Deviation (MAD), we need to find the absolute deviations from the mean, sum them, and then divide by the number of values. $\newline$ Absolute deviations: $|23 - 910.4|, |4 - 910.4|, |6 - 910.4|, |7 - 910.4|, |8999 - 910.4|, |9 - 910.4|, |10 - 910.4|, |13 - 910.4|, |15 - 910.4|, |18 - 910.4|$ $\newline$ Sum of absolute deviations: $887.4 + 906.4 + 904.4 + 903.4 + 8088.6 + 901.4 + 900.4 + 897.4 + 895.4 + 892.4 = 16177.2$ $\newline$ MAD = Sum of absolute deviations / $n$ = $16177.2 / 10 = 1617.72$
Compute First Quartile: To compute the first quartile $\theta_{1}$ , we need to find the median of the lower half of the data (excluding the median if $n$ is odd). Since our data set is even, we take the lower $5$ values. $\newline$ Lower half of the data: $4$ , $6$ , $7$ , $9$ , $10$ $\newline$ Median of the lower half $\theta_{1}$ = $7$ (the middle value of the lower half)
Compute IQR: To compute the Interquartile Range (IQR), we subtract the first quartile from the third quartile ( $Q_{3}$ ). $\newline$ First, we need to find the third quartile ( $Q_{3}$ ), which is the median of the upper half of the data. $\newline$ Upper half of the data: $13, 15, 18, 23, 8999$ $\newline$ Median of the upper half ( $Q_{3}$ ) = $18$ (the middle value of the upper half) $\newline$ Now, we can calculate the IQR. $\newline$ IQR = $Q_{3} - \theta_{1} = 18 - 7 = 11$
Compute Variance: To compute the variance, we need to find the squared deviations from the mean, sum them, and then divide by the number of values. $\newline$ Squared deviations: $(23 - 910.4)^2$ , $(4 - 910.4)^2$ , $(6 - 910.4)^2$ , $(7 - 910.4)^2$ , $(8999 - 910.4)^2$ , $(9 - 910.4)^2$ , $(10 - 910.4)^2$ , $(13 - 910.4)^2$ , $(15 - 910.4)^2$ , $(18 - 910.4)^2$ $\newline$ Sum of squared deviations: $(4 - 910.4)^2$ $0$ $\newline$ Variance = Sum of squared deviations / $(4 - 910.4)^2$ $1$ = $(4 - 910.4)^2$ $2$
Compute Standard Deviation: To compute the standard deviation, we take the square root of the variance. Standard deviation = $\sqrt{\text{variance}}$ = $\sqrt{6543996.65}$ $\approx 2558.12$