Use the display of data to find the standard deviation.
A bar graph titled Score with a horizontal axis labeled from 5 to 11 in increments of 1 and a vertical axis labeled Frequency from 0 to 12 in increments of 2 contains three vertical bars with labels and heights as follows, where the horizontal axis label is listed first and the height is listed second: 7, 12; 8, 8; 9, 12.
Score
5
6
7
8
9
10
11
Frequency
0
2
4
6
8
10
12
Neat looking
Ah, the bar graph of joy! Let's calculate that standard deviation, shall we?
First, we need to find the mean (average) of the scores. So, we add up all the scores and divide by the total number of scores. In this case, it looks like:
(7 x 12) + (8 x 8) + (9 x 12) = 132 + 64 + 108 = 304
We then divide this sum by the total number of scores (12 + 8 + 12 = 32):
Mean = 304 / 32 = 9.5
Next, we calculate the difference between each score and the mean, square it, and sum up all the squared differences. Let's calculate the squared differences for each score, shall we?
For the score 7: (7 - 9.5)^2 = (-2.5)^2 = 6.25
For the score 8: (8 - 9.5)^2 = (-1.5)^2 = 2.25
For the score 9: (9 - 9.5)^2 = (-0.5)^2 = 0.25
Next, we multiply each squared difference by its respective frequency:
(6.25 x 12) + (2.25 x 8) + (0.25 x 12) = 75 + 18 + 3 = 96
Now, we divide this sum by the total number of scores minus 1 (32 - 1 = 31) to get the variance:
Variance = 96 / 31 ≈ 3.0968
And finally, we take the square root of the variance to find the standard deviation:
Standard Deviation ≈ √3.0968 ≈ 1.7582
So, the standard deviation for this set of scores is approximately 1.7582.
Ta-da! I hope this helps, and remember, statistics don't have to be scary, but they can be a bit of a circus act! 🤡
To find the standard deviation from the given bar graph, follow these steps:
Step 1: Find the mean (average) of the scores using the bar graph.
To find the mean, you need to multiply each score by its respective frequency and then sum all the results. The formula for finding the mean is:
Mean = (Sum of (Score * Frequency)) / (Sum of Frequencies)
Calculating the sum of (Score * Frequency):
(7 * 12) + (8 * 8) + (9 * 12) = 84 + 64 + 108 = 256
Calculating the sum of frequencies:
12 + 8 + 12 = 32
Mean = 256 / 32 = 8
So, the mean score is 8.
Step 2: Find the squared differences from the mean.
For each score, subtract the mean from the score and square the result. The formula for finding the squared differences is:
Squared Difference = (Score - Mean) ^ 2
Calculating the squared differences:
(7 - 8) ^ 2 = 1
(8 - 8) ^ 2 = 0
(9 - 8) ^ 2 = 1
Step 3: Calculate the variance.
The variance is found by summing the squared differences and dividing it by the total frequency. The formula for calculating the variance is:
Variance = (Sum of Squared Differences) / (Total Frequency)
Calculating the sum of squared differences:
1 + 0 + 1 = 2
Total Frequency = 12 + 8 + 12 = 32
Variance = 2 / 32 = 0.0625
Step 4: Find the standard deviation.
The standard deviation is the square root of the variance. So, take the square root of the variance calculated in Step 3.
Standard Deviation = √Variance = √0.0625 = 0.25
Therefore, the standard deviation is 0.25.
To find the standard deviation, we need to calculate the mean and then subtract it from each value in the data set. Here's how you can do it:
Step 1: Calculate the mean (average):
The mean is calculated by adding up all the values and dividing by the total number of values. In this case, we have three bars with corresponding heights. Let's calculate the mean:
Mean = (7*12 + 8*8 + 9*12) / (12 + 8 + 12)
= (84 + 64 + 108) / 32
= 256 / 32
= 8
So, the mean is 8.
Step 2: Subtract the mean from each value:
Now, subtract the mean (8) from each bar's height:
7 - 8 = -1
8 - 8 = 0
9 - 8 = 1
Step 3: Square each result:
Next, square each of these results:
(-1)^2 = 1
0^2 = 0
1^2 = 1
Step 4: Calculate the mean of the squared values:
Find the mean of the squared values from step 3. In this case, we have 3 values, so the mean will be the sum of the squared values divided by 3:
Mean = (1 + 0 + 1) / 3
= 2 / 3
= 0.66666667
Step 5: Take the square root of the mean:
Finally, take the square root of the mean from step 4 to get the standard deviation:
Standard Deviation = √0.66666667
≈ 0.816496581
So, the standard deviation of the data set is approximately 0.816.
Find the mean first = sum of scores/number of scores
[(6*2)+ (7*4)...+(11*12)]/42 = ?
Subtract each of the scores from the mean and square each difference. Find the sum of these squares. Divide that by the number of scores to get variance.
Standard deviation = square root of variance
I'll let you do the calculations.