# Mode And Median – Statistical Model Exam

### Mode And Median, Statistical Model, Mode, Median, Distribution, Data Set – Exam

The key terms in these Statistic chapters include Mode And Median, Data Set, Ordinal, Interval, Mode, Statistical Model, Population, Ratio Scale, Variance, Mean, Sample, Nominal, Z-Score, Distribution, Median, Data, Standard Deviation, Sample Variance, Population Variance, Sample Mean, Math Exam.

Mode

Most frequently occurring category or score in distribution
Only one that can be used for nominal data
Benefits: easily calculated, no at all effected by extremes, can be determined graphically, usually actual value of an important part of series
Limitations: not capable of mathematical manipulation, affected by sampling fluctuations, insensitive to large changes in data set

Median

Score in the middle of data set
Measure of ordinal, interval, or ratio scale
(N+1)/2 if N is even
Used when distribution is skewed
Benefits: can be calculated always, relatively unaffected by extremes, located graphically, most useful dealing with qualitative data
Limitations: not capable of further manipulation, affected fluctuation of sampling, in case of even number, may not be a value from data

Mean

Requires interval or ratio and data cannot be skewed
sum set of score then divide by N
sensitive to every score

Nominal

##### The mode is only measure that can be used

Ordinal

Mode and median may be used

Interval/ration

All three may be used
mean provides most information
median preferred is distribution is skewed

Deterministic model

Always produce the same output from a give starting condition or initial state

Statistical model

Variable are predicted with some variation or uncertainty
includes both deterministic and random components

Only equation you’ll need for statistical model

Outcome = (model) + error

Why do we use a statistical model

To represent what is happening in the real world

Calculating “error”

Deviance (difference between mean and actual data point) = outcome – model
(take each score and subtract the mean from it)

Sum of Squared Errors

We square each deviation
so, SS = (each score and subtract mean from each and then square each score)

Sample variance

Average of the squared deviation of scores around the sample mean
Take each score and subtract mean from each
Then square each new score

Mode

##### Most frequently occurring category or score in distributionOnly one that can be used for nominal dataBenefits: easily calculated, no at all effected by extremes, can be determined graphically, usually actual value of an important part of seriesLimitations: not capable of mathematical manipulation, affected by sampling fluctuations, insensitive to large changes in data set

Median

Score in the middle of data set
Measure of ordinal, interval, or ratio scale
(N+1)/2 if N is even
Used when distribution is skewed
Benefits: can be calculated always, relatively unaffected by extremes, located graphically, most useful dealing with qualitative data
Limitations: not capable of further manipulation, affected fluctuation of sampling, in case of even number, may not be a value from data

Mean

Requires interval or ratio and data cannot be skewed
sum set of score then divide by N
sensitive to every score

Nominal

The mode is only measure that can be used

Ordinal

Mode and median may be used

Variability measures

##### Range, variance, SD

Variability

Describes how the scores are scattered around a central point

Range

Total distance covered by distribution
highest – lowest

Variance

Measure of how data points differ from the mean (center of distribution)
SD^2 = E(x-M)^2/N

A population of N = 10 scores has µ = 21 and σ = 3. What is the population variance?

#### 9

What the variance for the following sample of n = 3 scores? Scores: 1, 4, 7

#### 9

A researcher is interested in whether a new reading technique influences the reading ability of elementary school children. The researcher knows that average reading ability among a population of third graders is µ = 3 with a standard deviation of σ = 1. If the researcher administers the new reading technique to a third grader who subsequently scores X = 4, it can be concluded that the new reading technique _____.

Does not have an influence on reading ability.

The distribution of z-scores corresponding with a population of scores always has a variance of _____.

σ^2 = 1.

A person scores X = 65 on an exam. Which set of parameters would give this person the worst grade on the exam relative to others?

µ = 70 and σ = 5

Which z-score corresponds to a score that is above the mean by 2 standard deviations?

+2

Which of the following symbols identifies the population standard deviation?

#### σ

Sample vs population variance

Use n-1 for sample (degree of freedom)

z distribution

Always has a mean of 0 and SD of 1

Three properties of standard scores

1. The mean of a set of z-scores is always 0
2. SD is always = 1
3. The distribution os a set of standardized scores has the same shape as the unstandardized scores

Centile scores

Proportion of people with scores less than or equal to a particular score

z-score

Measure of observations distance from the mean and tells us where value fits into a normal distribution
if zero, on the mean
if +, above the mean
For -, below the mean
if 1, its 1 SD above mean
if -2, its 2 SDs below the mean

Standard score

Raw score expressed in terms of how many SDs it falls away form mean (or z score)

Formula for zscores

Population:
z = (X-weird u)/weird o (or (raw score – population mean)/sample SD)

Sample:
z = (X-M)/s (or (raw score – mean score)/SD of scores)

A sample of n = 4 scores, has Σ X = 4, and Σ X 2 = 32. What is the value of SS for this sample?

#### 28

Which of the following is true for most distributions?

Around 70% of the scores will be located within one standard deviation of the mean.

A sample of n = 20 scores has a mean of M = 32 and a standard deviation of s = 6. In this sample, what is the z-score corresponding to X = 28?

z = -.67

For a population with µ = 65 and σ = 4, what is the X value corresponding to z = -2.25?

X = 56

A population distribution has σ = 6. What position in this distribution is identified by a z-score of z = +2.33?

14 points above the mean

Last week Sarah had exams in her math and Spanish classes. On the math exam, the mean was µ = 40 with σ = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 8, and Sarah had a score of X = 68. For which class should Sarah expect the better grade relative to her peers in each class?

The grades should be the same because the two exam scores are in the same location.

A population of N = 10 scores has µ = 21 and σ = 3. What is the population variance?

#### 9

What the variance for the following sample of n = 3 scores? Scores: 1, 4, 7

#### 9

A researcher is interested in whether a new reading technique influences the reading ability of elementary school children. The researcher knows that average reading ability among a population of third graders is µ = 3 with a standard deviation of σ = 1. If the researcher administers the new reading technique to a third grader who subsequently scores X = 4, it can be concluded that the new reading technique _____.

Does not have an influence on reading ability.

The distribution of z-scores corresponding with a population of scores always has a variance of _____.

σ^2 = 1.

A person scores X = 65 on an exam. Which set of parameters would give this person the worst grade on the exam relative to others?

µ = 70 and σ = 5

Which z-score corresponds to a score that is above the mean by 2 standard deviations?

+2

Which of the following symbols identifies the population standard deviation?

#### σ

Mean

Requires interval or ratio and data cannot be skewed
sum set of score then divide by N
sensitive to every score

Nominal

The mode is only measure that can be used

Ordinal

Mode and median may be used

Variability measures

Range, variance, SD

Variability

Describes how the scores are scattered around a central point

Range

Total distance covered by distribution
highest – lowest

Last week Sarah had exams in her math and Spanish classes. On the math exam, the mean was µ = 40 with σ = 5, and Sarah had a score of X = 45. On the Spanish exam, the mean was µ = 60 with σ = 8, and Sarah had a score of X = 68. For which class should Sarah expect the better grade relative to her peers in each class?

The grades should be the same because the two exam scores are in the same location.

Formula for zscores

Population:
z = (X-weird u)/weird o (or (raw score – population mean)/sample SD)

Sample:
z = (X-M)/s (or (raw score – mean score)/SD of scores)

Sample variance

Average of the squared deviation of scores around the sample mean
Take each score and subtract mean from each
Then square each new score