Math 3680

Lecture #2

Mean and Standard Deviation

Mean vs. Median

Example: In a certain class of 13 students, 10 showed up the first exam, while 3 blew it off: Here are the grades; in order:

(A) Calculate the class median.

(i) Include all students.

(ii) Ignore the students who slept in.

(B) Calculate the class mean (average).

(i) Include all students.

(ii) Ignore the students who slept in.

98949387848179786855000

Definition: Sample mean. For a data set of size n, the sample mean is

n

i

i

n

xx

1

N

i

i

N

x

1

Definition: Population mean. For a finite population of size N, the population mean is

Example:

Suppose the student who got a 55 instead got a 15. Would the median change? Would the mean?

Example:

Suppose the 98 is replaced by 980. Would the median change? Would the mean? By how much?

Note: The mean is much more sensitive to wild outliers than the median.

98949387848179786855000

Exercise: For registered students at universities in the U.S., which is larger: average age or median age?

Repeat for the heights of 12-year-olds.

Repeat for the weights of 12-year-olds.

Repeat for the scores on a college final exam.

Like the median, the mean only captures central behavior and does not contain information about the spread of the data.

Physical interpretation of the mean: a “balance.”

Physical interpretation of the median: half the area lies on each side.

We have just explored the ideas of mean (average), median and mode. These measurements are useful in providing succinct numerical representations for measures of central tendencies.

Exercise:

Two different groups of 10 students are given identical quizzes with the following results. Compute the mean, median, and mode.

Group A 65 66 67 68 71 73 74 77 77 77

Group B 42 54 58 62 67 77 77 85 93 100

Standard Deviation

Definition: Sample Standard Deviation. For a data set of size n, the sample standard deviation is

n

ii xx

ns

1

2)(1

1

1.      Square all of the deviations from average.

2.      Sum the squares, then divide by n - 1 (the degrees of freedom).

3. Take the square root of the result of step 2.

Intuition: The standard deviation gives a measure of how “spread out” the data is.

Exercise: For each list below, find x and s:

(i) 1, 4, 6, 7, 8, 10

(ii) 5, 8, 10, 11, 12, 14

(iii) 3, 12, 18, 21, 24, 30

Example: Each of the following lists has an average of 50. For which one is the SD of the numbers the biggest? Smallest?

0, 20, 40, 50, 60, 80, 100

0, 48, 49, 50, 51, 52, 100

0, 1, 2, 50, 98, 99, 100

Example: For a list of positive numbers, can the SD ever be larger than the average?

70000 94780 105000 121391 135000 Average = 115,953.20$ 77028 95000 105194 122160 144416 SD = 26,810.27$ 85000 95000 106078 124575 14500085000 95000 107482 124855 14513285506 95389 110000 125130 15000085776 96361 110298 126485 15480090000 98331 115345 127303 17500093089 98500 117000 131768 17500093600 100600 120087 132500 17700094532 102704 120303 133162 179000

For large data sets, Microsoft Excel can compute the mean and standard deviation.

www.math.unt.edu/~allaart/3680/governors.xls

=AVERAGE(A1:E10) =STDEV(A1:E10)

1) The SD says how far away numbers on a list are from their average. Most entries on the list will be somewhere around one SD away from the average. Very few will be more than two or three SDs away.

2) Roughly 68% of the values will be within one SD of the average, and 95% will be within two SDs.(This is only a rule of thumb!)

70000 94780 105000 121391 135000 Average = 115,953.20$ 77028 95000 105194 122160 144416 SD = 26,810.27$ 85000 95000 106078 124575 14500085000 95000 107482 124855 145132 Average - 2 SD = 62,332.66$ 85506 95389 110000 125130 150000 Average - 1 SD = 89,142.93$ 85776 96361 110298 126485 154800 Average = 115,953.20$ 90000 98331 115345 127303 175000 Average + 1 SD = 142,763.47$ 93089 98500 117000 131768 175000 Average + 2 SD = 169,573.74$ 93600 100600 120087 132500 17700094532 102704 120303 133162 179000

Example: Estimate the mean of the high temperatures recorded in Denton over the past 30 days. Then estimate the standard deviation.

Definition: Population standard deviation.

N

ii xx

N 1

2)(1

This formula should be used in the (rare) occasion

that the entire population is known, not a sample.

Definition: Sample variance:

n

ii xx

ns

1

22 )(1

1

N

ii xx

N 1

22 )(1

Definition: Population variance:

Grouped Data

Grouped Data

To handle grouped data, we pretend that all members of each class are located at the midpoint (called the mark).

Find and for the age of the population under 50% of the poverty threshold.

Q: Why aren’t we finding x and s?

Q: Will our answer be exact?

0 18 9 556100018 25 21.5 250700025 35 30 215500035 45 40 179200045 55 50 154000055 60 57.5 61400060 65 62.5 53700065 85 75 932000

To handle grouped data, we pretend that all members of each class are located at the midpoint (called the mark).

Now compute the mean and standard deviation:

Definition: Grouped mean:

where m = number of groups

m

iii xxf

ns

1

22 )(1

1

m

iii xxf

N 1

22 )(1

Definition: Grouped Population variance:

Definition: Grouped sample variance:

m

iii xf

nx

1

1