### 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****median provides more information**

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****Add all of them together**

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****median provides more information**

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 distributionif zero, on the meanif +, above the meanFor -, below the meanif 1, its 1 SD above meanif -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****median provides more information**

Variability measures

**Range, variance, SD**

Variability

**Describes how the scores are scattered around a central point**

Range

**Total distance covered by distributionhighest – 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 meanTake each score and subtract mean from eachThen square each new scoreAdd all of them together**