Mean, Standard Deviation, and Normal Distribution - Algebra 2

To begin, we must remember the formula for standard deviation:

where is the standard deviation, N is the number of values in our set, is the value we're currently evaluating in the summation, and is the mean of our set of numbers. All the summation part of the equation means is that we subtract our mean from each number in the set, square that value, and then add all of those values for each number together. So before we find the values that will be added together, we must first find our mean for the set of numbers:

Now we can determine the value for our summation for each number in the set:

Looking at our equation for standard variation, now all we must do is sum all of the values above, divide by N, and take the square root:

Two standard deviations means to double the standard deviation value.

This means the range is to .

Only is not in the range.

where n equals the number of events.

To answer this question, you must first calculate Willow's total points from the 3 exams. The total points will be equal to 3 times the original average:

95*3=285

Now, if the 90 is removed, his new total will be:

285-90=195

With the score dropped, her new total number of exams is 2. Therefore, the new average is:

This question is really just asking for the average dollar per page. This could be easily calculated:

However, you might also think of this as a rate problem:

D=RT

Using the information from the question, we can see that D=5000 and T=2200.

This comes out to what we solved above:

This question relates to the 68−95−99.7 rule of normal distribution. We know that 95% of the data are within 2 standard deviations from the mean.

In this case this means that 95% of the students are between

15.5−2⋅0.6 and 15.5+2⋅0.5

15−1.2 and 15+1.2

13.8 and 16.2

Therefore we have 5% of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 13.8 is the same as the proportion of students above 16.2.

Thus the right answer is one half of 5%, or 2.5%.

2.5% of the students are older than 16.2 years old.

This question relates to the 68−95−99.7 rule of normal distribution. We know that 95% of the data are within 2 standard deviations from the mean.

In this case this means that 95% of the students are between

23.5−2⋅2.5 and 23.5+2⋅2.5

23.5-5 and 23.5+5

18.5 and 28.5

Therefore we have 5% of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 18.5 is the same as the proportion of students above 28.5.

Thus the right answer is half of 5% or 2.5%.

By drawing a bell curve, the middle line is 50%. One standard deviation left and right of the middle line is 34% each. That means one standard deviation within is 68%.

Another way to calculate standard deviation is the square root of variance.

Variance is,

.

Taking the square root of this is how standard deviation can be calculated.

The first step in solving for standrd deviation is to find the mean of the data set.

For this problem:

.

Now we can evaluate the summation:

Now we can rewrite the standard deviation expression:

There are 5 data points, so n = 5

To begin, we must remember the formula for standard deviation:

where is the standard deviation, N is the number of values in our set, is the value we're currently evaluating in the summation, and is the mean of our set of numbers. All the summation part of the equation means is that we subtract our mean from each number in the set, square that value, and then add all of those values for each number together. So before we find the values that will be added together, we must first find our mean for the set of numbers:

Now we can determine the value for our summation for each number in the set:

Looking at our equation for standard variation, now all we must do is sum all of the values above, divide by N, and take the square root:

Two standard deviations means to double the standard deviation value.

This means the range is to .

Only is not in the range.

where n equals the number of events.

To answer this question, you must first calculate Willow's total points from the 3 exams. The total points will be equal to 3 times the original average:

95*3=285

Now, if the 90 is removed, his new total will be:

285-90=195

With the score dropped, her new total number of exams is 2. Therefore, the new average is:

This question is really just asking for the average dollar per page. This could be easily calculated:

However, you might also think of this as a rate problem:

D=RT

Using the information from the question, we can see that D=5000 and T=2200.

This comes out to what we solved above:

This question relates to the 68−95−99.7 rule of normal distribution. We know that 95% of the data are within 2 standard deviations from the mean.

In this case this means that 95% of the students are between

15.5−2⋅0.6 and 15.5+2⋅0.5

15−1.2 and 15+1.2

13.8 and 16.2

Therefore we have 5% of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 13.8 is the same as the proportion of students above 16.2.

Thus the right answer is one half of 5%, or 2.5%.

2.5% of the students are older than 16.2 years old.

This question relates to the 68−95−99.7 rule of normal distribution. We know that 95% of the data are within 2 standard deviations from the mean.

In this case this means that 95% of the students are between

23.5−2⋅2.5 and 23.5+2⋅2.5

23.5-5 and 23.5+5

18.5 and 28.5

Therefore we have 5% of the students that are outside of this range. Since the normal distribution is symmetric, the proportion of students below 18.5 is the same as the proportion of students above 28.5.

Thus the right answer is half of 5% or 2.5%.

By drawing a bell curve, the middle line is 50%. One standard deviation left and right of the middle line is 34% each. That means one standard deviation within is 68%.

Another way to calculate standard deviation is the square root of variance.

Variance is,

.

Taking the square root of this is how standard deviation can be calculated.