Statistical Reasoning

Statistical ReasoningA. Describing data

1. Frequency distributions – Where are the majority of the scores?

13 A+ 40 13 A+12 A 39 4 41% A 12 23 52% 12 A 11 39%

38 11 A- 11 10 11 A- 1411 A- 37 11 B+ 10 15 10 B+ 910 B+ 36 6 B 9 5 41% 9 B 12 45%9 B 35 4 31% B- 8 6 8 B- 8

34 5 C+ 7 2 7 C+ 28 B- 33 5 C 6 1 5% 6 C+ 2 11%7 C+ 32 4 C- 5 5 C- 36 C 31 3 19% D+ 4 2 4 D+

30 3 D 3 3% 3 D 3 5%5 C- 29 2 D- 2 2 D-4 D+ 28 F 1 0% 1 F 0%3 D 27 4 8% 0

26 12 D- 251 F 24 2%

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Multiple Choice

Essay

Composite

Mean=34.3SD=4.2

Mean=10.2SD=2.0

Mean=9.3SD=2.3

Statistical Reasoning

A. Describing data

1. Frequency distributions

2. Histograms & frequency polygons – ways of showing your frequency distribution data.

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A B C D

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HistogramUses a Bar Graph to show data

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Frequency PolygonUses a line graph to show data

Statistical ReasoningB. Measures of central tendency – 3 types

4, 3, 5, 4, 41. Mode=most common=4(Reports what there is more of – Used in data with no

connection. Can’t average men & women.)

2. Mean=arithmetic average=20/5=4(has most statistical value)

3. Median=middle score=4(1/2 the scores are higher, half are lower.

Used when there are extreme scores)

Central Tendency An extremely high or low price/score can skew the mean. Sometimes the

median is better at showing you the central tendency.

1968 TOPPS Baseball CardsNolan Ryan $1500Billy Williams $8Luis Aparicio $5Harmon Killebrew $5Orlando Cepeda $3.50Maury Wills $3.50Jim Bunning $3Tony Conigliaro $3Tony Oliva $3Lou Pinella $3Mickey Lolich $2.50

Elston Howard $2.25Jim Bouton $2Rocky Colavito $2Boog Powell $2Luis Tiant $2Tim McCarver $1.75Tug McGraw $1.75Joe Torre $1.5Rusty Staub $1.25Curt Flood $1

With Ryan:Median=$2.50Mean=$74.14

Without Ryan:Median=$2.38Mean=$2.85

Does the mean accurately portray the central tendency of incomes?

NO!

What measure of central tendency would more accurately show income distribution?

Median – the majority of the incomes surround that number.

Statistical ReasoningA. Describing dataB. Measures of central tendencyC. Measures of variation 1. Range – Difference b/w a high & low score (can be skewed by an extreme score)

2. Variance & standard deviation – How spread out is your data?

Calculating Standard DeviationHow spread out (consistent) is your data?

1. Calculate the mean.

2. Take each score and subtract the mean from it.

3. Square the new scores to make them positive.

4. Mean (average) the new scores

5. Take the square root of the mean to get back to your original measurement.

6. The smaller the number the more closely packed the data. The larger the number the more spread out it is.

Standard Deviation

PuntDistance

36384145

Mean:160/4 = 40 yds

Deviationfrom Mean

36 - 40 = -438 – 40 = -241 – 40 = +145 – 40 = +5

DeviationSquared

Numbers multiplied by itself & added together

16 4 125

46Variance:

46/4 = 11.5

StandardDeviation:

variance=

11.5 = 3.4 yds

Z-ScoresA number expressed in Standard Deviation Units that shows

an Individual score’s deviation from the mean.Basically, it shows how you did compared to everyone else.

+ Z-score means you are above the mean, – Z-score means you are below the mean.

Z-Score = your score minus the average score divided by standard deviation.

Which class did you perform better in compared to your classmates?

Test Total Your Score

Average score

S.D.

Biology 200 168 160 4

Psych. 100 44 38 2

Z score in Biology: 168-160 = 8, 8 / 4 = +2 S.D.

Z score in Psych: 44-38 = 6, 6/2 = +3 S.D.

You performed better in Psych compared to your classmates.

Statistical ReasoningA. Describing data

B. Measures of central tendency

C. Measures of variation

D. Characteristics of the normal curve

Skewed Curves

A Positive Skew has a tail that goes to the right.

A Negative Skew has a tail that goes to the left.

Statistical ReasoningA. Describing data

B. Measures of central tendency

C. Measures of variation

D. Characteristics of the normal curve

E. Inference

1. Does the sample represent the population?

a. Non-biased sample-good

b. Low variability-good

c. Larger samples-good

Statistical ReasoningE. Inference

1. Does the sample represent the pop.? 2. Are differences between groups in your

results statistically significant?

a. Big differences-good

b. Low variability-good

c. Big groups-good

Statistical Significancep value = likelihood a result is caused by chance• This is bad to a researcher. They want this number to be as

small as possible to show that any change in their experiment was caused by an independent variable and not some outside force.

• This number can be no greater than 5% for the findings to be considered statistically significant.p ≤ .05

• This means the researcher must be 95% certain their results are not caused by chance.

• Replication of the experiment will prove the p value to be true or not.