Question

jordyn asked 8 people how many apps they each had on their phone. here are their responses: 21, 28, 31, 46, 55, 60, 65, 70 the mean is \\( \bar{x} = 47 \\) apps. which of these formulas gives the standard deviation? choose 1 answer: \\( \boldsymbol{\text{a}} \\) \\( s_x = \sqrt{\frac{(21 - 8)^2 + (28 - 8)^2 + \cdots + (70 - 8)^2}{47}} \\)

Explanation:

To determine the correct formula for the standard deviation, we recall the formula for the sample standard deviation:

The formula for the sample standard deviation \( s_x \) is:

\[
s_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
\]

where \( x_i \) are the individual data points, \( \bar{x} \) is the sample mean, and \( n \) is the number of data points.

In this problem, we have:

• Data points: \( 21, 28, 31, 46, 55, 60, 65, 70 \)
• Sample mean \( \bar{x} = 47 \)
• Number of data points \( n = 8 \)

Let's analyze the given option (A) and the correct formula:

Option A:

\[
s_x = \sqrt{\frac{(21 - 8)^2 + (28 - 8)^2 + \cdots + (70 - 8)^2}{47}}
\]

This is incorrect because:

• The deviation is taken from \( 8 \) (which is the number of data points, not the mean \( \bar{x} = 47 \)).
• The denominator is \( 47 \) (the mean), not \( n - 1 = 7 \) or \( n = 8 \).

Correct Formula:

The correct formula for the sample standard deviation should be:

\[
s_x = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + (31 - 47)^2 + (46 - 47)^2 + (55 - 47)^2 + (60 - 47)^2 + (65 - 47)^2 + (70 - 47)^2}{8 - 1}}
\]

Simplifying the deviations:

• \( 21 - 47 = -26 \), so \( (21 - 47)^2 = (-26)^2 = 676 \)
• \( 28 - 47 = -19 \), so \( (28 - 47)^2 = (-19)^2 = 361 \)
• \( 31 - 47 = -16 \), so \( (31 - 47)^2 = (-16)^2 = 256 \)
• \( 46 - 47 = -1 \), so \( (46 - 47)^2 = (-1)^2 = 1 \)
• \( 55 - 47 = 8 \), so \( (55 - 47)^2 = 8^2 = 64 \)
• \( 60 - 47 = 13 \), so \( (60 - 47)^2 = 13^2 = 169 \)
• \( 65 - 47 = 18 \), so \( (65 - 47)^2 = 18^2 = 324 \)
• \( 70 - 47 = 23 \), so \( (70 - 47)^2 = 23^2 = 529 \)

Summing these squared deviations:
\[
676 + 361 + 256 + 1 + 64 + 169 + 324 + 529
\]

Calculating the sum:
\[
676 + 361 = 1037 \\
1037 + 256 = 1293 \\
1293 + 1 = 1294 \\
1294 + 64 = 1358 \\
1358 + 169 = 1527 \\
1527 + 324 = 1851 \\
1851 + 529 = 2380
\]

Then, divide by \( n - 1 = 7 \):
\[
\frac{2380}{7} = 340
\]

Finally, take the square root:
\[
s_x = \sqrt{340} \approx 18.44
\]

But since the question is about identifying the correct formula (not calculating the numerical value), the key points are:

• The deviation is from the mean \( \bar{x} = 47 \), not from \( n = 8 \).
• The denominator is \( n - 1 = 7 \) (for sample standard deviation) or \( n = 8 \) (for population standard deviation). However, since we have a sample (Jordyn asked 8 people, which is a sample from a larger population), we use \( n - 1 \).

Since the provided option (A) is incorrect, and assuming there are other options (not fully shown in the image), the correct formula should involve deviations from the mean \( 47 \) and a denominator of \( 7 \) (or \( 8 \) if it's population standard deviation).

If we assume the problem is about sample standard deviation, the correct formula is:

\[
s_x = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + \cdots + (70 - 47)^2}{8 - 1}}
\]

If it's about population standard deviation, the denominator would be \( 8 \):

\[
\sigma = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + \cdots + (70 - 47)^2}{8}}
\]

Since the mean is given as \( \bar{x} = 47 \), the correct formula must have \( (x_i - 47) \) in the numerator.

Answer:

To determine the correct formula for the standard deviation, we recall the formula for the sample standard deviation:

The formula for the sample standard deviation \( s_x \) is:

\[
s_x = \sqrt{\frac{\sum_{i=1}^{n} (x_i - \bar{x})^2}{n - 1}}
\]

where \( x_i \) are the individual data points, \( \bar{x} \) is the sample mean, and \( n \) is the number of data points.

In this problem, we have:

• Data points: \( 21, 28, 31, 46, 55, 60, 65, 70 \)
• Sample mean \( \bar{x} = 47 \)
• Number of data points \( n = 8 \)

Let's analyze the given option (A) and the correct formula:

Option A:

\[
s_x = \sqrt{\frac{(21 - 8)^2 + (28 - 8)^2 + \cdots + (70 - 8)^2}{47}}
\]

This is incorrect because:

• The deviation is taken from \( 8 \) (which is the number of data points, not the mean \( \bar{x} = 47 \)).
• The denominator is \( 47 \) (the mean), not \( n - 1 = 7 \) or \( n = 8 \).

Correct Formula:

The correct formula for the sample standard deviation should be:

\[
s_x = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + (31 - 47)^2 + (46 - 47)^2 + (55 - 47)^2 + (60 - 47)^2 + (65 - 47)^2 + (70 - 47)^2}{8 - 1}}
\]

Simplifying the deviations:

• \( 21 - 47 = -26 \), so \( (21 - 47)^2 = (-26)^2 = 676 \)
• \( 28 - 47 = -19 \), so \( (28 - 47)^2 = (-19)^2 = 361 \)
• \( 31 - 47 = -16 \), so \( (31 - 47)^2 = (-16)^2 = 256 \)
• \( 46 - 47 = -1 \), so \( (46 - 47)^2 = (-1)^2 = 1 \)
• \( 55 - 47 = 8 \), so \( (55 - 47)^2 = 8^2 = 64 \)
• \( 60 - 47 = 13 \), so \( (60 - 47)^2 = 13^2 = 169 \)
• \( 65 - 47 = 18 \), so \( (65 - 47)^2 = 18^2 = 324 \)
• \( 70 - 47 = 23 \), so \( (70 - 47)^2 = 23^2 = 529 \)

Summing these squared deviations:
\[
676 + 361 + 256 + 1 + 64 + 169 + 324 + 529
\]

Calculating the sum:
\[
676 + 361 = 1037 \\
1037 + 256 = 1293 \\
1293 + 1 = 1294 \\
1294 + 64 = 1358 \\
1358 + 169 = 1527 \\
1527 + 324 = 1851 \\
1851 + 529 = 2380
\]

Then, divide by \( n - 1 = 7 \):
\[
\frac{2380}{7} = 340
\]

Finally, take the square root:
\[
s_x = \sqrt{340} \approx 18.44
\]

But since the question is about identifying the correct formula (not calculating the numerical value), the key points are:

• The deviation is from the mean \( \bar{x} = 47 \), not from \( n = 8 \).
• The denominator is \( n - 1 = 7 \) (for sample standard deviation) or \( n = 8 \) (for population standard deviation). However, since we have a sample (Jordyn asked 8 people, which is a sample from a larger population), we use \( n - 1 \).

Since the provided option (A) is incorrect, and assuming there are other options (not fully shown in the image), the correct formula should involve deviations from the mean \( 47 \) and a denominator of \( 7 \) (or \( 8 \) if it's population standard deviation).

If we assume the problem is about sample standard deviation, the correct formula is:

\[
s_x = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + \cdots + (70 - 47)^2}{8 - 1}}
\]

If it's about population standard deviation, the denominator would be \( 8 \):

\[
\sigma = \sqrt{\frac{(21 - 47)^2 + (28 - 47)^2 + \cdots + (70 - 47)^2}{8}}
\]

Since the mean is given as \( \bar{x} = 47 \), the correct formula must have \( (x_i - 47) \) in the numerator.