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A sample data set contains the following seven values:

6
2
4
9
1
3
5
7
10
8

(a) Find the following measures of central tendency
(i) Mode
(ii) Mean
(iii) Median
(b) Find the following measures of dispersion
(i) range.
(ii) mean absolute deviation.
(iii) variance.
(iv) interquartile range.

$3$ . A sample data set contains the following seven values: $\newline$ \begin{tabular}{llllllllll} $\newline$ \hline $6$ & $2$ & $4$ & $9$ & $1$ & $3$ & $5$ & $7$ & $10$ & $8$ \\ $\newline$ \hline $\newline$ \end{tabular} $\newline$ (a) Find the following measures of central tendency $\newline$ (i) Mode $\newline$ (ii) Mean $\newline$ (iii) Median $\newline$ (b) Find the following measures of dispersion $\newline$ (i) range. $\newline$ (ii) mean absolute deviation. $\newline$ (iii) variance. $\newline$ (iv) interquartile range.

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Math Problems

Algebra 1

Calculate quartiles and interquartile range

Full solution

3

. A sample data set contains the following seven values:

\newline

\begin{tabular}{llllllllll}

\newline

\hline

6

2

4

9

1

3

5

7

10

8

\newline

\hline

\newline

\end{tabular}

\newline

(a) Find the following measures of central tendency

\newline

(i) Mode

\newline

(ii) Mean

\newline

(iii) Median

\newline

(b) Find the following measures of dispersion

\newline

(i) range.

\newline

(ii) mean absolute deviation.

\newline

(iii) variance.

\newline

(iv) interquartile range.

Sort Data Set: Step $1$ : Sort the data set in ascending order for easier calculations. $\newline$ Data: $1, 2, 3, 4, 5, 6, 7, 8, 9, 10$
Find Mode: Step $2$ : Find the mode. $\newline$ Mode: No number repeats, so no mode.
Calculate Mean: Step $3$ : Calculate the mean (average). $\newline$ Mean = $(1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10) / 10 = 55 / 10 = 5.5$
Determine Median: Step $4$ : Determine the median (middle value). $\newline$ Median: Since there are $10$ numbers, median = average of $5$ th and $6$ th numbers = $(5 + 6) / 2 = 5.5$
Calculate Range: Step $5$ : Calculate the range (difference between highest and lowest). $\newline$ Range = $10 - 1 = 9$
Mean Absolute Deviation: Step $6$ : Calculate the mean absolute deviation. $\newline$ Mean absolute deviation = $(|1-5.5| + |2-5.5| + |3-5.5| + |4-5.5| + |5-5.5| + |6-5.5| + |7-5.5| + |8-5.5| + |9-5.5| + |10-5.5|) / 10$ $\newline$ = $(4.5 + 3.5 + 2.5 + 1.5 + 0.5 + 0.5 + 1.5 + 2.5 + 3.5 + 4.5) / 10 = 25 / 10 = 2.5$
Calculate Variance: Step $7$ : Calculate the variance. $\newline$ Variance = $[(1-5.5)^2 + (2-5.5)^2 + (3-5.5)^2 + (4-5.5)^2 + (5-5.5)^2 + (6-5.5)^2 + (7-5.5)^2 + (8-5.5)^2 + (9-5.5)^2 + (10-5.5)^2] / 10$ $\newline$ = $[20.25 + 12.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25 + 12.25 + 20.25] / 10 = 82.5 / 10 = 8.25$
Calculate IQR: Step $8$ : Calculate the interquartile range (IQR). $\newline$ $IQR = Q3 - Q1$ $\newline$ $Q1 =$ median of first $5$ numbers $= 3$ $\newline$ $Q3 =$ median of last $5$ numbers $= 8$ $\newline$ $IQR = 8 - 3 = 5$