Question

grace is looking at a report of her monthly cell - phone usage for the last year to determine if she needs to upgrade her plan. the list represents the approximate number of megabytes of data grace used each month: 700, 735, 680, 890, 755, 740, 670, 785, 805, 1050, 820, 750. what is the standard deviation of the data? round to the nearest whole number. 65 75 100 130

Explanation:

Step1: Find the mean ($\mu$) of the data set.

The data set is: 700, 735, 680, 890, 755, 740, 670, 785, 805, 1050, 820, 750 (assuming 12 data points for a year, maybe a typo earlier, let's confirm: 700,735,680,890,755,740,670,785,805,1050,820,750 – 12 values).
Sum of data: $700 + 735 + 680 + 890 + 755 + 740 + 670 + 785 + 805 + 1050 + 820 + 750$
Calculate sum:
$700+735 = 1435$; $1435+680 = 2115$; $2115+890 = 3005$; $3005+755 = 3760$; $3760+740 = 4500$; $4500+670 = 5170$; $5170+785 = 5955$; $5955+805 = 6760$; $6760+1050 = 7810$; $7810+820 = 8630$; $8630+750 = 9380$
Mean $\mu=\frac{9380}{12}\approx781.67$

Step2: Calculate each deviation from the mean, square it.

For each $x_i$:

• $700 - 781.67 = -81.67$; $(-81.67)^2\approx6670.0$
• $735 - 781.67 = -46.67$; $(-46.67)^2\approx2178.0$
• $680 - 781.67 = -101.67$; $(-101.67)^2\approx10337.0$
• $890 - 781.67 = 108.33$; $(108.33)^2\approx11735.0$
• $755 - 781.67 = -26.67$; $(-26.67)^2\approx711.0$
• $740 - 781.67 = -41.67$; $(-41.67)^2\approx1736.0$
• $670 - 781.67 = -111.67$; $(-111.67)^2\approx12463.0$
• $785 - 781.67 = 3.33$; $(3.33)^2\approx11.1$
• $805 - 781.67 = 23.33$; $(23.33)^2\approx544.3$
• $1050 - 781.67 = 268.33$; $(268.33)^2\approx71991.0$
• $820 - 781.67 = 38.33$; $(38.33)^2\approx1469.2$
• $750 - 781.67 = -31.67$; $(-31.67)^2\approx1003.0$

Step3: Sum the squared deviations.

Sum of squared deviations: $6670 + 2178 + 10337 + 11735 + 711 + 1736 + 12463 + 11.1 + 544.3 + 71991 + 1469.2 + 1003$
Calculate step by step:
$6670+2178=8848$; $8848+10337=19185$; $19185+11735=30920$; $30920+711=31631$; $31631+1736=33367$; $33367+12463=45830$; $45830+11.1=45841.1$; $45841.1+544.3=46385.4$; $46385.4+71991=118376.4$; $118376.4+1469.2=119845.6$; $119845.6+1003=120848.6$

Step4: Calculate variance ($\sigma^2$) and standard deviation ($\sigma$).

Variance: $\sigma^2=\frac{120848.6}{12}\approx10070.72$
Standard deviation: $\sigma=\sqrt{10070.72}\approx100.35\approx100$ (rounded to nearest whole number)

Answer:

100