Mean Stats

STATS

not as calm and peaceful as this picture

STATISTICS

COVERED:

Mean, median, mode

Normal Distribution Curve

Standard Deviation

Z score

Margin of Error

​Mean

The mean is just the average:

Find the mean of the set {3,4,5}

Add the numbers together 3+4+5=12

Divide by the number of items (3), 12÷3=4

Therefore, the mean is 4.

(In Desmos, the formula is mean(x₁)

In Excel, the formula is =Average(highlight cells)

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Determine the mean of the following data set:

{80, 82, 65, 78, 69, 72, 79}

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

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Determine the mean of the following data set:

{63, 58, 69, 55, 49, 64, 70, 52}

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

MEDIAN

The one in the middle

The median is the value in the middle of the set when the set is in order from least to greatest.

Find the median of the set {3,1,2}

Put the values in order {1,2,3}

2 is the value in the middle, it is the mean

In Desmos, the formula is median(x₁)

In Excel, the formula is =Median(highlight cells)

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Find the median of the data set:

{60, 52, 11, 29, 46, 9, 33}

(If necessary round to the nearest tenth, i.e. 2.85=2.9)

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Find the median of the data set:

{5, 14, 22, 35, 20, 15, 6, 28}

(If necessary round to the nearest tenth, i.e. 2.85=2.9)

MODE

The one in the middle

The mode is the value that appears the most in a data set.

Find the mode of the set {24, 31, 12, 38, 12, 15}

12 appears twice so 12 is the mode

If a value does not appear more than others, is has no mode.

It is possible to have more than one mode.

In Desmos, there is no formula for mode, but dot plot works!

In Excel, the formula is =Mode.Mult(highlight cells)

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Determine the mode of the data set:

{5, 28, 16, 32, 5, 16, 48, 29, 5, 35}

(If there is more than one mode, put them in order from least to greatest separated by a comma, i.e. 1,2,3)

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Determine the mode of the data set:

{53, 13, 34, 41, 26, 61, 34, 13, 69}

(If there is more than one mode, put them in order from least to greatest separated by a comma, i.e. 1,2,3)

STANDARD DEVIATION

Average absolute distance from the mean

Find the standard deviation of the set {20 21 19 22 21 20 19 20 21 20}

Standard deviation = 0.95

In Desmos, the formula is stdevp(x₁)

In Excel, the formula is =stdev.p(highlight cells)

It is possible to calculate standard deviation by hand, but it is long and involved - remember to bring your computer!

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Find the standard deviation of the data set:

{303 299 306 298 304 307 299 302 305 299 300}

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

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Find the standard deviation of the data set:

{87 89 84 88 89 87 86 87 86 87}

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

NORMAL DISTRIBUTION

​Notice, opposite sides of the curve add up to 100.

The mean is in the middle.

Add the calculated standard deviation to move up.

Subtract the standard deviation to move down.

For example:

The mean is 25.5 and the standard distribution is 3.5

What is the value at 2 sd (standard distributions) above the mean?

25.5+3.5=29 (1SD added)

29+3.5=32.5 (2SD added)

The value at 2SD above the mean is 32.5

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A data set with a normal distribution has a mean of 50 and a standard deviation of 10. What value is 1SD below the mean?

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

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A data set with a normal distribution has a mean of 68 and a standard deviation of 7.8. What value is 2SD above the mean?

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

What percent of the data is from -1SD to 2SD of the mean?

-1SD is 16%

2SD is 97.5%

Subtract the smallest value from the largest value.

97.5-16=81.5

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A data set has a normal distribution curve, what percentage of the data is within -2SD and 3SD of the mean?

(If necessary, round to the nearest hundredth, 2.865=2.87)

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A data set has a normal distribution curve, what percentage of the data is not within -3SD and 1SD of the mean?

(If necessary, round to the nearest hundredth, 2.865=2.87)

x is the value

is simply the number of standard deviations a value is above or below the mean on a normal distribution curve.

Z-score

Z-Score

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A normal distribution curve has a mean of 75 and a standard distribution of 12.5

Calculate the z-score if the value is 48

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

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A university will only admit students who scores 2SD above the mean on the ACT. The mean on the ACT is 18 and has a standard deviation of 6.

What is the z-score of someone who scored a 32 on the ACT?

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

​Solving for a value

If given the z-score, mean, and standard deviation, the value may be calculate by manipulating the equation:

value=(z-score)×(standard deviation)+mean

For example:

A normal distribution curve has a standard deviation of 2.5 and a mean of 86. Calculate the value for a z-score of 1.5

value = (1.5)×(2.5)+86

value = 89.75

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A normal distribution curve has mean of 125 and a standard distribution of 12.5.

Calculate the value for a z-score of -1.25

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

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A normal distribution curve has mean of 78.3 and a standard distribution of 8.4.

Calculate the value for a z-score of 2.2

(If necessary, round to the nearest tenth, i.e. 2.85=2.9)

MARGIN OF ERROR

The margin of error (MOE) is range added/subtracted to a response percentage based on the number of people surveyed. This is to allow for variations in opinions do to not all people being surveyed.

The formula is MOE is 1/√n

Where n is the number of people surveyed.

For example, 80% of people of 400 surveyed disagreed with a policy. What is the margin of error? 1/√400 =0.05 so the MOE is ±5%

80-5=75, 80+5=85 therefore the range is 75-85

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A survey was conducted of 10,000 students. 25.5% of the students indicated they liked homework. Calculate the margin of error.

(Just the margin of error, not the range.)

(Do not include the % in your answer)

(If necessary, round to the nearest tenth, 2.85=2.9)

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A survey was conducted of 1250 students. 75.8% of the students indicated they loved Mr. Salomon's math class. Calculate the margin of error and indicate the upper end of the range

(i.e. if the range is 10-15, 15 is the upper end of the range)

(Do not include the % in your answer)

(If necessary, round to the nearest tenth, 2.85=2.9)

MARGIN OF ERROR

If you manipulate the margin of error formula, it is possible to determine how many people are surveyed.

If MOE=1/√n then n= (1/MOE)²

For example, how people must be surveyed to achieve an MOE of 5.5%?

n=(1/0.055)²

n=330.6 but remember these are people so n=330

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How many people must be surveyed in order to obtain a margin of error of 2.75%?

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How many people must be surveyed in order to obtain a margin of error of 1.25%?