q14. using the data values given, answer parts a through e.
5 5 5 5 12 6 2 2 2 2 7 9 11 8 7
a. find the mean.
b. find the median.
c. find the mode.
d. find the standard deviation.
e. determine the percentile rank of 2.

Question

Accepted Answer

### Part a: Find the mean
# Explanation:
## Step1: Count the number of data points
There are 15 data points: $5, 5, 5, 5, 12, 6, 2, 2, 2, 2, 7, 9, 11, 8, 7$
## Step2: Calculate the sum of the data
$$
\begin{align*}
\text{Sum} &= 5\times4 + 12 + 6 + 2\times4 + 7\times2 + 9 + 11 + 8\
&= 20 + 12 + 6 + 8 + 14 + 9 + 11 + 8\
&= 20+12=32; 32+6=38; 38+8=46; 46+14=60; 60+9=69; 69+11=80; 80+8=88
\end{align*}
$$
## Step3: Calculate the mean (Mean = Sum / Number of data points)
$$
\text{Mean} = \frac{88}{15} \approx 5.87
$$
# Answer:
$\approx 5.87$

### Part b: Find the median
# Explanation:
## Step1: Arrange the data in ascending order
$2, 2, 2, 2, 5, 5, 5, 5, 6, 7, 7, 8, 9, 11, 12$
## Step2: Find the middle value (since there are 15 data points, the 8th value is the median)
The 8th value is $5$
# Answer:
$5$

### Part c: Find the mode
# Explanation:
## Step1: Identify the value(s) with the highest frequency
- $2$ appears 4 times.
- $5$ appears 4 times.
- Other values have lower frequencies.
## Step2: Determine the mode(s)
Both $2$ and $5$ are modes as they appear most frequently (4 times each).
# Answer:
$2$ and $5$

### Part d: Find the standard deviation
# Explanation:
## Step1: Recall the formula for sample standard deviation: $s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}$ (where $\bar{x}$ is the mean, $x_i$ are data points, $n$ is the number of data points)
We know $\bar{x}=\frac{88}{15}\approx5.87$, $n = 15$
## Step2: Calculate $(x_i - \bar{x})^2$ for each data point and sum them up
$$
\begin{align*}
&\text{For } x = 2: (2 - \frac{88}{15})^2 = (\frac{30 - 88}{15})^2 = (\frac{-58}{15})^2=\frac{3364}{225}\text{ (4 times)}\
&\text{For } x = 5: (5 - \frac{88}{15})^2 = (\frac{75 - 88}{15})^2 = (\frac{-13}{15})^2=\frac{169}{225}\text{ (4 times)}\
&\text{For } x = 6: (6 - \frac{88}{15})^2 = (\frac{90 - 88}{15})^2 = (\frac{2}{15})^2=\frac{4}{225}\
&\text{For } x = 7: (7 - \frac{88}{15})^2 = (\frac{105 - 88}{15})^2 = (\frac{17}{15})^2=\frac{289}{225}\text{ (2 times)}\
&\text{For } x = 8: (8 - \frac{88}{15})^2 = (\frac{120 - 88}{15})^2 = (\frac{32}{15})^2=\frac{1024}{225}\
&\text{For } x = 9: (9 - \frac{88}{15})^2 = (\frac{135 - 88}{15})^2 = (\frac{47}{15})^2=\frac{2209}{225}\
&\text{For } x = 11: (11 - \frac{88}{15})^2 = (\frac{165 - 88}{15})^2 = (\frac{77}{15})^2=\frac{5929}{225}\
&\text{For } x = 12: (12 - \frac{88}{15})^2 = (\frac{180 - 88}{15})^2 = (\frac{92}{15})^2=\frac{8464}{225}\
\end{align*}
$$
## Step3: Calculate the sum of squared deviations
$$
\begin{align*}
\text{Sum of squared deviations}&= 4\times\frac{3364}{225}+ 4\times\frac{169}{225}+\frac{4}{225}+ 2\times\frac{289}{225}+\frac{1024}{225}+\frac{2209}{225}+\frac{5929}{225}+\frac{8464}{225}\
&=\frac{13456 + 676 + 4 + 578 + 1024 + 2209 + 5929 + 8464}{225}\
&=\frac{13456+676=14132; 14132+4=14136; 14136+578=14714; 14714+1024=15738; 15738+2209=17947; 17947+5929=23876; 23876+8464=32340}{225}\
&=\frac{32340}{225}= 143.733\cdots
\end{align*}
$$
## Step4: Calculate the variance (Variance = Sum of squared deviations / (n - 1))
$$
\text{Variance} = \frac{32340/225}{14}=\frac{32340}{225\times14}=\frac{32340}{3150}\approx10.27
$$
## Step5: Calculate the standard deviation (Standard Deviation = $\sqrt{\text{Variance}}$)
$$
\text{Standard Deviation} = \sqrt{10.27} \approx 3.20
$$
# Answer:
$\approx 3.20$

### Part e: Determine the percentile rank of 2
# Explanation:
## Step1: Recall the formula for percentile rank: $\text{Percentile Rank} = \frac{\text{Number of values less than } x + 0.5\times\text{Number of values equal to } x}{n}\times100$
Here, $x = 2$, $n = 15$
## Step2: Count the number of values less than 2 (0) and equal to 2 (4)
## Step3: Calculate the percentile rank
$$
\text{Percentile Rank} = \frac{0 + 0.5\times4}{15}\times100=\frac{2}{15}\times100\approx13.33
$$
# Answer:
$\approx 13.33$

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QUESTION IMAGE

Question

Explanation:

Part a: Find the mean

Step1: Count the number of data points

Step2: Calculate the sum of the data

Step3: Calculate the mean (Mean = Sum / Number of data points)

Step1: Arrange the data in ascending order

Step2: Find the middle value (since there are 15 data points, the 8th value is the median)

Step1: Identify the value(s) with the highest frequency

Step2: Determine the mode(s)

Answer:

Part b: Find the median