pred 35 4 teach. probility & statis. for primary math

PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Lesson 3Central Tendency & Variability 

PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Q1. A researcher examined the effect of amount of relaxation

training on insomnia. Four treatment groups were used.

Subjects received relaxation training for 2, 4, or 8

sessions. A control group received no training (0

sessions). Following training, the researcher measured

how long it took the subjects to fall asleep. The average

time for each group is presented in the following table:

PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Q1.

a. Identify the IV and DV for this study.

Training sessions

Average time (in minutes)

0 72

2 58

4 33

8 14

b. What is scale of the measurement was used for the IV and the DV?

c. If the researcher used a graph to show the obtained relationship between the IV and the DV, what kind of graph would be appropriate? Sketch the graph showing the results of this experiment?

PRED 354 TEACH. PROBILITY & STATIS. FOR PRIMARY MATH

Q2. For the following set of the scores 

4, 6, 9, 5, 3, 8, 9, 4, 2, 5, 10, 7, 4 ,9, 8, 3

a. Construct a frequency distribution table

b. Sketch a polygon showing the distribution

c. Describe the shape of the distribution

d. What is the percentile rank for X=6?

e. What is the 70th percentile?

Central tendency

is a statistical measure that identifies a single score as representative of an entire distribution.

MEANMEDIANMODE

The Mean

The mean for a distribution is the sum of the scores divided by the number scores.

Quiz score (x) f

10 1

9 2

8 4

7 0

6 1

Charactistics of the Mean

1. Changing a score or introducing a new score.

2. Adding or subtracting a constant from each score.3. Multiplying or dividing each score by a constant.

The Median

is the score that divides a distribution exactly in half.

Three types of data?1. When N is odd number 2. When N is even number3. When there are several scores with the same

value in the middle of the distribution.

The Median

EX:

1. 3, 5, 8, 10, 11

2. 3, 3, 4, 5, 7, 8

3. 1, 2, 2, 3, 4, 4, 4, 4, 4, 5

EX: Find the median for this data

3, 4, 3, 2, 1, 3, 2, 4

The Mode

In a frequency distribution, the mode is the score or category that has greatest frequency.

Quiz score (x) f

5 2

4 6

3 4

2 2

1 1

How do you decide which measure of central tendency to use?

When to use mode?It can be used with any scale

of measurement.

Academic major f

Biology 2

Psyhics 6

Sociology 2

Mathematic 1

How do you decide which measure of central tendency to use?

When to use median?

1. There are a few extreme scores in the distribution

2. Some scores have undetermined values

3. There are open ended distribution

4. The data measured on an ordinal scale

How do you decide which measure of central tendency to use?

When to use median?

1. There are a few extreme scores in the distribution

Errors committed before reaching learning criterion (x)

f

10 1

11 4

12 3

13 1

100 1

How do you decide which measure of central tendency to use?

When to use median?

1. Some scores have undetermined values

Person (x) Time (amount of time to complete puzzle)

1 8

2 11

3 12

4 13

5 17

6 Never finished

How do you decide which measure of central tendency to use?

When to use median?

1. There are open ended distribution

Number of children (x)

f

5 or more

3

4 2

3 2

2 3

1 6

0 4

Central tendency and the shape of the distribution

1. Symmetrical distribution

2. Skewed distributions

Central tendency

EX:

Find the mode, median and mean

X f

5 10

4 6

3 2

2 1

1 1

Variability

provides a quantitative measure of the degree to which scores in a distribution are spread out or clustered together.

A. RANGE – B. SEMI-INTERQUARTILE RANGE C. STANDARD DEVIATION

PURPOSE: Are the scores all clustered together, or are they scattered over a wide range of values?

Range

The range is the distance between the largest score and the smallest score in the distribution.

range = URL Xmax – LRL Xmin

Ex. 3, 7, 12, 8, 5, 10

The Interquartile range and semi-interquartile range

The interquartile range is the distance between the first quartile and the third quartile.

interquartile range = Q3 – Q1

semi-interquartile range= ½ * (Q3 – Q1)

Ex. 2, 3, 4, 4, 5, 5, 6, 6, 6, 7, 7, 8, 8, 9, 10, 11

Standard deviation and variance for a population (a measure of distance from the mean)

Step 1. The first step in finding the standard distance from the mean is to determine the deviation for each individual score.

Deviation is the distance from the mean

Deviation score = X - µ

Standard deviation and variance for a population

Step 2. The next step is to calculate the mean of the deviation scores.

Step 3. Use mean squared deviation (Variance)

Population Variance = Mean squared deviation = SS/N

(SS = Σ (X - µ)2 )

Standard deviation and variance for a population

Step 4. Simply make a correction for having squared all the distances.

standard deviation = √variance

σ = √SS/N

σ2 = SS/N

x

1

0

6

1

EX:

Standard deviation and variance for a population

Use the following population of scores to calculate SS, variance, and standard deviation

Scores: 1, 9, 5, 8, 7

EX:

Standard deviation and variance for a sample

Notations: Notice that these sample formula use n-1 instead of n.

Population Sample

Mean µ X

variance σ2 = SS/N s2=SS/n-1

Standard deviation σ = √SS/N s = √SS/n-1

Standard deviation and variance for a sample

Why do we use n-1 for the sample?

Sample variability tends to underestimate population variability unless some correction is made.

Dividing by a smaller value produces a larger result and makes sample variability an accurate, or unbiased, estimator of population variability.

Standard deviation and variance for a sample

Degrees of freedom:n-1

df = n-1

s2=SS/df

s = √SS/df

Properties of SD

1. descriptive measure: distance from the mean

2. a measure of how big the error will be.

Properties of SD

1. Adding a constant to each score will not change the SD.

2. Multiplying each score by a constant causes the SD to be multiplied by the same constant.

SD or Range? When?

1. Extreme scores.

2. Sample size.

3. Stability under sampling

4. Open-ended distributions

SD or Range? When?

Range Semi-interquartile range

Variance SD

Extreme scores most least Be careful Be careful

Sample size Directly related

Relatively unaffected

Relatively unaffected

Relatively unaffected

Stability under sampling

unstable Reasonably stable

stable stable

Open-ended distributions

N/A Only available one

N/A N/A