The following data set represents a recent set of test scores in a Chemistry Class.
90, 90, 95, 100, 80, 80, 75, 80, 70, 60, 95, 100, 100, 100, 75, 80, 90, 90, 90, 70, 70, 80, 85, 90, 90, 85
Use a statistics calculator (Links to an external site.) to find the mean and standard deviation (round to nearest tenth) of the data set.
Mean:
Sample Standard Deviation:
Sketch out the following diagram to help you solve the problem and label the mean and three standard deviations from the mean in both directions.
Use the diagram above, and the mean and standard deviation you calculated to answer the following questions using the empirical rule.
a) How many students in the chemistry class (meaning how many of the 26 students) scored within 1 Standard Deviation of the mean? How does this compare to what the Empirical Rule says should happen?
b) How many students in the chemistry class (meaning how many of the 26 students) scored within 2 Standard Deviation of the mean? How does this compare to what the Empirical Rule says should happen?
Cannot diagram on these posts. Cannot copy and paste here.
To solve this problem, we will first calculate the mean and standard deviation of the given data set. Then, we will use the Empirical Rule to answer the questions.
Step 1: Calculating the mean and standard deviation
To find the mean, we will sum up all the test scores and divide by the number of students (26 in this case).
Mean = (90 + 90 + 95 + 100 + 80 + 80 + 75 + 80 + 70 + 60 + 95 + 100 + 100 + 100 + 75 + 80 + 90 + 90 + 90 + 70 + 70 + 80 + 85 + 90 + 90 + 85) / 26
Mean ≈ 84.4 (rounded to nearest tenth)
To find the standard deviation, we will use the statistics calculator (Links to an external site.) provided:
Sample Standard Deviation ≈ 11.7 (rounded to nearest tenth)
Step 2: Sketch the diagram
MEAN
|--------|--------|--------|--------|
-1σ -2σ +2σ +1σ
Step 3: Applying the Empirical Rule
a) We will calculate the range within 1 standard deviation of the mean.
Lower Bound: Mean - 1σ ≈ 84.4 - 11.7 ≈ 72.7
Upper Bound: Mean + 1σ ≈ 84.4 + 11.7 ≈ 96.1
Now, we count the number of students who scored within this range:
90, 90, 95, 80, 80, 75, 80, 70, 95, 80
There are 10 students who scored within 1 standard deviation of the mean.
According to the Empirical Rule, approximately 68% of the data should fall within 1 standard deviation of the mean. In this case, 10 out of 26 students fall within that range, which is close to the expected percentage.
b) We will calculate the range within 2 standard deviations of the mean.
Lower Bound: Mean - 2σ ≈ 84.4 - 2(11.7) ≈ 61.0
Upper Bound: Mean + 2σ ≈ 84.4 + 2(11.7) ≈ 107.8
Now, we count the number of students who scored within this range:
90, 90, 95, 100, 80, 80, 75, 80, 70, 60, 95, 100, 100, 100, 75, 80, 90, 90, 90, 70, 70, 80, 85, 90, 90, 85
There are 25 students who scored within 2 standard deviations of the mean.
According to the Empirical Rule, approximately 95% of data should fall within 2 standard deviations of the mean. In this case, 25 out of 26 students fall within that range, which is very close to the expected percentage.
To find the mean and standard deviation of the data set, you can use a statistics calculator.
1. Input the data set:
90, 90, 95, 100, 80, 80, 75, 80, 70, 60, 95, 100, 100, 100, 75, 80, 90, 90, 90, 70, 70, 80, 85, 90, 90, 85
2. Calculate the mean:
The mean is the sum of all the values divided by the total number of values. In this case, there are 26 values.
Mean = (90 + 90 + 95 + 100 + 80 + 80 + 75 + 80 + 70 + 60 + 95 + 100 + 100 + 100 + 75 + 80 + 90 + 90 + 90 + 70 + 70 + 80 + 85 + 90 + 90 + 85) / 26
Mean = 85
3. Calculate the sample standard deviation:
The sample standard deviation measures the average distance between each data point and the mean. You can use the formula:
Sample Standard Deviation = √(Σ(xi - x̄)^2 / (n-1))
where xi represents each data point, x̄ is the mean, and n is the sample size.
Sample Standard Deviation = √( (90 - 85)^2 + (90 - 85)^2 + ... + (85 - 85)^2 ) / (26-1)
After calculating the sum of each squared difference, you divide it by (n-1), and finally take the square root.
The result will be the sample standard deviation.
Now, to answer the questions using the empirical rule:
a) According to the empirical rule, approximately 68% of the data should fall within 1 standard deviation of the mean. To find out how many students scored within 1 standard deviation of the mean, you can calculate the range of values using the mean and 1 standard deviation in both directions.
Lower bound = mean - 1 standard deviation
Upper bound = mean + 1 standard deviation
Once you have the lower and upper bounds, count how many values in the data set fall within that range.
b) According to the empirical rule, approximately 95% of the data should fall within 2 standard deviations of the mean. To find out how many students scored within 2 standard deviations of the mean, you can calculate the range of values using the mean and 2 standard deviations in both directions.
Lower bound = mean - 2 standard deviations
Upper bound = mean + 2 standard deviations
Similar to the previous question, count how many values in the data set fall within that range.
By using the diagram provided, you can visualize the range of values within each standard deviation level from the mean.