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Math  /  Data & Statistics

QuestionThe stock prices for eight major grocery store chains last January were: $18.24$20.34$9.36$11.53$11.21$48.04$48.82$28.27\begin{array}{llllllll} \$ 18.24 & \$ 20.34 & \$ 9.36 & \$ 11.53 & \$ 11.21 & \$ 48.04 & \$ 48.82 & \$ 28.27 \end{array} Send data to Excel
Find the range, variance, and standard deviation. Round the variance to one decimal place and the standard deviation to two decimal places, if necessary.
Part 1 of 3
The range is $39.46\$ 39.46.
Part: 1/31 / 3
Part 2 of 3
The variance is □\square

Studdy Solution

STEP 1

What is this asking? We're looking at how spread out the grocery store stock prices were last January, using range, variance, and standard deviation. Watch out! Don't mix up variance and standard deviation!
Variance is the average of the squared differences from the mean, while standard deviation is the square root of the variance, representing a more interpretable measure of spread.

STEP 2

1. Calculate the Mean
2. Calculate the Variance
3. Calculate the Standard Deviation

STEP 3

Alright, let's **kick things off** by finding the **average** stock price!
We **add up** all the prices and then **divide** by the number of prices, which is **8**.

STEP 4

Mean=$18.24+$20.34+$9.36+$11.53+$11.21+$48.04+$48.82+$28.278\text{Mean} = \frac{\$18.24 + \$20.34 + \$9.36 + \$11.53 + \$11.21 + \$48.04 + \$48.82 + \$28.27}{8}

STEP 5

Mean=$185.818=$23.22625\text{Mean} = \frac{\$185.81}{8} = \$23.22625

STEP 6

So, the **mean** stock price is $23.22625\$23.22625.
We'll use this in our next exciting step!

STEP 7

Now, let's **calculate the variance**!
This tells us how much the individual stock prices typically *vary* from the mean.
We'll find the **squared difference** between each price and the mean, **add** those up, and then **divide** by the number of prices.

STEP 8

Variance=∑i=18(xi−Mean)28\text{Variance} = \frac{\sum_{i=1}^{8} (x_i - \text{Mean})^2}{8}

STEP 9

Let's **break it down**: Variance=18[($18.24−$23.22625)2+($20.34−$23.22625)2+($9.36−$23.22625)2+($11.53−$23.22625)2+($11.21−$23.22625)2+($48.04−$23.22625)2+($48.82−$23.22625)2+($28.27−$23.22625)2]\begin{aligned} \text{Variance} = \frac{1}{8} [ &(\$18.24 - \$23.22625)^2 + (\$20.34 - \$23.22625)^2 + (\$9.36 - \$23.22625)^2 \\ &+ (\$11.53 - \$23.22625)^2 + (\$11.21 - \$23.22625)^2 + (\$48.04 - \$23.22625)^2 \\ &+ (\$48.82 - \$23.22625)^2 + (\$28.27 - \$23.22625)^2] \end{aligned}

STEP 10

(−4.98625)2+(−2.88625)2+(−13.86625)2+(−11.69625)2+(−12.01625)2+(24.81375)2+(25.59375)2+(5.04375)2(-4.98625)^2 + (-2.88625)^2 + (-13.86625)^2 + (-11.69625)^2 + (-12.01625)^2 + (24.81375)^2 + (25.59375)^2 + (5.04375)^2

STEP 11

Variance=24.8626+8.3306+192.2776+136.8026+144.3876+615.7056+654.9956+25.44068\text{Variance} = \frac{24.8626 + 8.3306 + 192.2776 + 136.8026 + 144.3876 + 615.7056 + 654.9956 + 25.4406}{8}

STEP 12

Variance=1802.80288≈225.35035≈225.4\text{Variance} = \frac{1802.8028}{8} \approx 225.35035 \approx 225.4

STEP 13

So, the **variance** is approximately 225.4225.4.
Almost there!

STEP 14

Finally, the **standard deviation**!
This is simply the **square root** of the variance.

STEP 15

Standard Deviation=Variance=225.35035≈15.01\text{Standard Deviation} = \sqrt{\text{Variance}} = \sqrt{225.35035} \approx 15.01

STEP 16

The **standard deviation** is approximately 15.0115.01.
We did it!

STEP 17

The range is $39.46\$39.46.
The variance is approximately 225.4225.4.
The standard deviation is approximately 15.0115.01.

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