question #10
find the standard deviation of the following set of sample data:
2075 2055 3105 3380 2860
3840 3615 2180 3055 2965
3035 3685 3315 3160 3940
3190
220.05
417.31
581.19
187.67

Question

Accepted Answer

# Explanation:
## Step1: Calculate the mean
First, find the sum of the data:
$$
\begin{align*}
&2075 + 2055+3105+3380+2860+3840+3615+2180+3055+2965+3035+3685+3315+3160+3940+3190\
=&(2075 + 2055)+(3105+3380)+(2860+3840)+(3615+2180)+(3055+2965)+(3035+3685)+(3315+3160)+(3940+3190)\
=&4130+6485+6700+5795+6020+6720+6475+7130\
=&(4130+6485)+(6700+5795)+(6020+6720)+(6475+7130)\
=&10615+12495+12740+13605\
=&(10615+12495)+(12740+13605)\
=&23110+26345\
=&49455
\end{align*}
$$
There are $n = 16$ data points. The mean $\bar{x}=\frac{49455}{16}=3090.9375$

## Step2: Calculate the squared differences
For each data point $x_i$, calculate $(x_i-\bar{x})^2$
$$
\begin{align*}
(2075 - 3090.9375)^2&=(-1015.9375)^2 = 1032069.140625\
(2055-3090.9375)^2&=(-1035.9375)^2=1073167.91015625\
(3105 - 3090.9375)^2&=(14.0625)^2 = 197.75390625\
(3380-3090.9375)^2&=(289.0625)^2=83557.51953125\
(2860 - 3090.9375)^2&=(-230.9375)^2=53332.390625\
(3840-3090.9375)^2&=(749.0625)^2 = 561193.359375\
(3615-3090.9375)^2&=(524.0625)^2=274641.6015625\
(2180 - 3090.9375)^2&=(-910.9375)^2=829707.91015625\
(3055-3090.9375)^2&=(-35.9375)^2 = 1291.50390625\
(2965-3090.9375)^2&=(-125.9375)^2=15859.75390625\
(3035-3090.9375)^2&=(-55.9375)^2=3129.00390625\
(3685-3090.9375)^2&=(594.0625)^2=352910.75390625\
(3315-3090.9375)^2&=(224.0625)^2=50194.25390625\
(3160-3090.9375)^2&=(69.0625)^2=4769.6015625\
(3940-3090.9375)^2&=(849.0625)^2=721007.359375\
(3190-3090.9375)^2&=(99.0625)^2=9813.390625
\end{align*}
$$
Sum of squared differences $\sum_{i = 1}^{n}(x_i-\bar{x})^2=1032069.140625+1073167.91015625+197.75390625+83557.51953125+53332.390625+561193.359375+274641.6015625+829707.91015625+1291.50390625+15859.75390625+3129.00390625+352910.75390625+50194.25390625+4769.6015625+721007.359375+9813.390625 = 5453939.9375$

## Step3: Calculate the sample variance
The sample variance $s^2=\frac{\sum_{i = 1}^{n}(x_i-\bar{x})^2}{n - 1}=\frac{5453939.9375}{15}\approx363595.995833$

## Step4: Calculate the sample standard deviation
The sample standard deviation $s=\sqrt{s^2}=\sqrt{363595.995833}\approx  581.19$

# Answer:
581.19

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the mean

Step2: Calculate the squared differences

Step3: Calculate the sample variance

Step4: Calculate the sample standard deviation

Answer: