Descriptive Statistics. My immediate family includes my wife Barbara, my sons Adam and Devon, and...

Descriptive Statistics

My immediate family includes my wife Barbara, my sons Adam and Devon, and myself. I am 62, Barbara is 61, and the boys are both 30. Barbara and Devon have master’s degrees, Adam has a bachelor’s degree, and I have a doctorate.

Descriptive Variables

• Name• Age• Gender• Education• Relationship to the family

Why Use Statistics

• It is not easy to describe all the characteristics (lots of variables) of all of the members of a group (lots of people).

• Summaries of the characteristics of the members of a group are called descriptive statistics.

Problems with Gathering Information About Characteristics

• Did you get the same information from each respondent?

• Is the information appropriate to your problem? • Can you transform the information into numbers?• Are the numbers in a form that can be analyzed?

The solution to these problems is to use measurement scales.

A Quick ReviewMeasures of Central Tendency

• Mode—the response that occurs most frequently

• Median—the point where half of the scores are above and half below

• Mean—average

Mode

23334456678

The response that occurs most frequently

Median

23334456678

The point at which half the scores are higher and half are lower

Mean

23334456678

4.64The average

Nominal Scales

• Unordered classification– Think of this as a group of containers into which you

will sort data.• Allows comparison of group sizes– Which container has the most in it?

• No information is embedded in the order of the categories

• Mode (the only measure of central tendency)

What color is your car?Nominal—Mode

Ordinal Scales• Ordered classification– Containers where it makes sense that they are in order

• Allows comparison of both group sizes and relative position of categories

• Categories are ordered but not evenly spaced– Some containers may be larger or smaller than others– The distance between the containers may not be equal

• Median (best measure of central tendency) or Mode

Do you like chocolate?Ordinal—Median or Mode

Interval Scales

• Ordered Classification– Just like ordinal the order makes sense

• Categories are ordered and evenly spaced– Unlike ordinal all of the containers are of equal size

and spaced evenly• Mean (best measure of central tendency)• Ratio scales are the same as interval except they

start at zero

How many ham sandwiches did you eat last week?

Ratio/Interval—Mean, Median or Mode

Descriptive Variables

• Name• Age• Gender• Education• Relationship to the family

Think up a nominal, ordinal and interval scale related to each of the following:

• Political affiliation• Restaurant ratings• Temperature• Shoe size• Teaching assignments• Teacher effectiveness• Income

Test Scores

• 27• 28• 28• 27• 27• 24• 27

• 26• 20• 30• 23• 23• 24

First: What kind of variable is Test Score?

Measures of Central Tendency

• Mode—the response that occurs most frequently

• Median—the point where half of the scores are above and half below

• Mean—average

Computing Measures of C T

• Lay out all of the scores in numerical order• Compute the mode by finding the number

that occurs most often• Compute the median by finding the middle

number in the list of scores• Compute the mean by adding up all of the

numbers and dividing by the number of numbers

Computing Measures of C T

20 23 23 24 24 26 27 27 27 27 28 28 30

Mode(most frequent)

Median(midpoint of responses)

Mean = 344/13 or 25.69(average)

Frequency Distribution

• The number of scores at each possible level20 — 121 — 022 — 023 — 224 — 225 — 026 — 127 — 428 — 229 — 030 — 1

20 23 23 24 24 26 27 27 27 27 28 28 30

HistogramBar chart of a frequency distribution

20 — 121 — 022 — 023 — 224 — 225 — 026 — 127 — 428 — 229 — 030 — 1

Score

Frequency

HistogramBar chart of a frequency distribution

20 — 121 — 022 — 023 — 224 — 225 — 026 — 127 — 428 — 229 — 030 — 1

Score

Frequency

ModeMedian

Mean 25.69

Histogram

Exercise—On a piece of paper:1. Make a histogram2. Compute measures of

central tendency

22222323252627272727282829

Histogram

Mode = 27Median = 27Mean = 25.69

Histogram



Measures of Variability

• Range—the distance between the highest and lowest score

• Standard Deviation—the average distance all the scores are from the mean

• Well kind of…

Standard Deviation

Computing Standard Deviation

• X = mean• n = each score• N = total number of scores• ∑ = sum (in this case, the sum of the differences

of each score from the mean, squared)

∑(X-n)2

N-1Standard Deviation =

Standard Deviation

20 22 24 26 28 30

20232324242627272727282830

X = 25.69

5.692.692.691.691.69

-.31-1.31-1.31-1.31-1.31-2.31-2.31-4.31

∑(X-n)2

N-1

Standard Deviation20 (5.69)23 (2.69)23 (2.69)24 (1.69)24 (1.69)26 (-.31)27 (-1.31)27 (-1.31)27 (-1.31)27 (-1.31)28 (-2.31)28 (-2.31)30 (-4.31)

∑(X-n)2

N-1x (5.69)x (2.69)x (2.69)x (1.69)x (1.69)x (-.31)x (-1.31)x (-1.31)x (-1.31)x (-1.31)x (-2.31)x (-2.31)x (-4.31)

Square each difference tomake them positive


∑(X-n)2

N-1x (5.69)x (2.69)x (2.69)x (1.69)x (1.69)x (-.31)x (-1.31)x (-1.31)x (-1.31)x (-1.31)x (-2.31)x (-2.31)x (-4.31)

= 32.4= 7.25= 7.25= 2.86= 2.86= 0.09= 1.71= 1.71= 1.71= 1.71= 5.33= 5.33= 18.6

Squared differencesfrom the mean


∑(X-n)2

N-1x (5.69)x (2.69)x (2.69)x (1.69)x (1.69)x (-.31)x (-1.31)x (-1.31)x (-1.31)x (-1.31)x (-2.31)x (-2.31)x (-4.31)

= 32.4= 7.25= 7.25= 2.86= 2.86= 0.09= 1.71= 1.71= 1.71= 1.71= 5.33= 5.33= 18.6

88.77 Sum of squared differences


∑(X-n)2

N-1x (5.69)x (2.69)x (2.69)x (1.69)x (1.69)x (-.31)x (-1.31)x (-1.31)x (-1.31)x (-1.31)x (-2.31)x (-2.31)x (-4.31)

= 32.4= 7.25= 7.25= 2.86= 2.86= 0.09= 1.71= 1.71= 1.71= 1.71= 5.33= 5.33= 18.6

88.77

Average of squared differences

88.77 / (13-1) = 7.40


∑(X-n)2

N-1x (5.69)x (2.69)x (2.69)x (1.69)x (1.69)x (-.31)x (-1.31)x (-1.31)x (-1.31)x (-1.31)x (-2.31)x (-2.31)x (-4.31)

= 32.4= 7.25= 7.25= 2.86= 2.86= 0.09= 1.71= 1.71= 1.71= 1.71= 5.33= 5.33= 18.6

88.77

Average of squared differences

88.77 / (13-1) = 7.40

Average of differences

7.40 = 2.72

Histogram

Mode = 27Median = 27Mean = 25.69Range = 10SD = 2.72

Mode = 27Median = 27Mean = 25.69Range = 7SD = 2.43

475492555

1. Compute the mean2. Subtract each score from the mean (9 scores—9 differences)3. Square each difference4. Add up the squares5. Divide by n-1 (8)6. Compute the square root

Compute the Standard Deviation

∑(X-n)2

N-1

Descriptive Statistics. My immediate family includes my wife Barbara, my sons Adam and Devon, and...

Documents

Transcript of Descriptive Statistics. My immediate family includes my wife Barbara, my sons Adam and Devon, and...