Statistics Chapter 4 and 5

Z score equation

(x-mean)/standard deviation

Two formulas for SS

1.Defintional Formula find each deviation score (x-u) square each deviation score (x-u)2 Sum up the squared deviations SS=E(x-u)2 2.Computational Formula Square each score and sum the squared scores find the sum of scores, square it, divide by N subtract the second part from the first SS=EX2-(Ex)2/N

Define Standard Deviation

1.Determine the deviation (distance from the mean-x-u) 2.Find the sum of deviations to use as a basis of finding an average deviation. 2.Revised-remove negative deviations-first square each deviations score, then sum the squared deviations 3.Average the squared deviations-mean squared deviation is known as the variance--variability is now measured in squared units Population variance equals mean squared deviation of the scores from the population mean 4.Goal to compute a measure of the standard distance of the scores from the mean--variance measures the average squared distance from the mean -not quite the goal---adjust for having squared all the differences by taking the square root of the variance.

Three measures of Variability

1.Range 2.The Variance 3.THe standard Deviation

Goal for defining and measuring variability

A measure of variability describes the degree to which the scores in a distribution are spread out or clustered together. Variability also measures the size of the distances between scores.

Describe the scores in a sample that has a standard deviation of zero.

A standard deviation of zero indicates there is no variability. In this case, all of the scores in the sample have exactly the same value.

Variance easy

Average Squared Deviation

Adding a constant to each score

DOES NOT CHANGE THE STANDARD DEVIATION

Measure of Variability Purpose

Describe the distribution Measure how well an individual score represents the distribution

Characteristics of a Z score

Every x value can be transformed to a Z score Mean of Z score is always 0 Standard Deviation is always 1.00 Z score distribution is called a standardized distribution

If all the scores in a data set are the same, the standard deviation is equal to 1.0

False-

The computational and definitional formulas for SS sometimes give different results

False-the results are identical

If a sample of n=10 scores is transformed into z scores, there will be five positive z scores and five negative z scores

False.....the number of Z scores above/below m will be exactly the same as the number of original scores above/below the mean.

A score close to the mean has a z score close to 1.00?

False....close to 0 is the mean

Which of the following explains why it is necessary to make a correction to the formula for sample variance?

If sample variance is computed by dividing by n, instead of n − 1, the resulting values will tend to underestimate the population variance.

Why would you want to transform a set of raw scores into a set of z scores?

Make it possible to compare scores from 2 different distributions. To be able to describe the location of a raw scores in distribution of raw scores.

Range

Measures the distance covered by the scores in a distribution--from smallest to highest For continuous data, real limits are used For discrete data, the maximum score and minimum score are used

Standard Deviation

Most common and most important measure of variability Measure the standard, or average, distance from the mean Describes whether the scores are clustered closely around the mean or are widely scattered Always have three standard deviations of mean.

Is it possible to obtain a negative value for variance or standard deviation?

No, they cannot be negative. Standard deviation is squared--no---and variance is sum of squares so no.

Computing Z scores for a sample

Relative position indicator Indicates distance from sample mean

Sample Standard Deviation symbol

S

Population Variance formula

SS/N

Sample Variance Formula

SS/n-1

The standard deviation measures...

Standard distance of a score from the mean

There are two different formulas or methods that can be used to calculate SS. Under what circumstances is the definitional formula easy to use? Under what circumstances is the computational formula preferred?

The definitional formula works well when the mean is a whole number and there is a relatively small number of scores. b. The computational formula is better when the mean is a fraction or decimal value and usually easier with a large number of scores.

A consequence of increasing variability is

The distance from one score to another tends to increase and a single score tends to provide a less accurate representation of the entire distribution.

Explain how a z-score identifies an exact location in a distribution with a single number.

The sign of the z-score tells whether the location is above (+) or below (-) the mean, and the magnitude tells the distance from the mean in terms of the number of standard deviations.

A negative z score always indicates a location below the mean

True

Transforming an entire distribution of scores into Z scores will not change the shape of the distribution

True

Population variance formula

Variance = sum of squared deviations/number of scores SS is the sum of the squared deviations of scores from the mean Two formulas for computing SS--defintional and computational

Variance

Variance equals the mean of the squared deviations. Variance is the average squared distance from the mean.

Comparisons of Z scores

When standardized, all Z scores are comparable Z scores allow direct comparisons from different distributions because they have been converted on the same scale

Under what circumstances is the computational formula preferred over the definitional formula when computing SS, the sum of the squared deviations, for a sample?

When the sample mean is not a whole number

A z score of z= +1.00 indicates a position in a distribution

above the mean by a distance equal to 1 standard deviation

For the following scores which action will increase the range? 3, 7, 10, and 15

add four points to x = 15

Standardized Distribution

composition of data used to make dissimilar distributions comparable

Line Graph

diagram used when values on horizontal axis are measured on an interval or ratio scale

Bimodal

distribution with two scores with greatest frequency

Inferential Statistics

drawing conclusions about a population based on sample data from that population Samples differ from the population-less variability--computing variance and standard deviation in the same way as for a population would give a biased estimate of the population values.

A sample statistic is unbiased

f the average value of the statistic is equal to the population parameter. (The average value of the statistic is obtained from all the possible samples for a specific sample size, n.)

A sample statistic is biased

if the average value of the statistic either underestimates or overestimates the corresponding population parameter.

Median

midpoint in a list of scores listed in order from smallest to largest

Degrees of Freedom

n-1

Biased

on the average, it consistently overestimates or underestimates the corresponding population parameter.

Raw score

original, unchanged datum that is the direct result of measurement

Variability

provides a quantitative measure of the differences between scores in a distribution and describes the degree to which the scores are spread out or clustered together. Quantitative distance--differences between scores (range) Distance of the spread of scores or the distance of a score from the mean (standard deviation) Measures predictability, consistency, and diversity

Z score transformation

relabeling of X values in a population into precise X-value locations within a distribution

Standardized Score

result from relabeling data into new table with positive, whole-number predetermined mean and standard deviation

Deviation

s the difference between a score and the mean, and is calculated as: deviation = X - μ

Mode

score or category that has the greatest frequency in a frequency distribution

Minor Mode

shorter peak when two scores with greatest frequency have unequal frequencies

Z score

specification of the precise location of each X value within a distribution -Exact location above or below the mean -distance between score and mean in standard deviation

Population Standard Deviation Formula

square root of SS/N

Central Tendancy

statistical measure to determine a single score that defines the midpoint of a distribution

Sample Variance symbol

s²

Sample Variance

s² = (SS / n-1)

Major mode

taller peak when two scores with greatest frequency have unequal frequencies

Sample Standard Deviation

the square root of the sample variance

Standard Deviation easy

the square root of the variance

Multiplying each score by a constant causes

the standard deviation to be multiplied by the same constant

How to determine a raw score from a z score

x= mean + zO

Population Standard Deviation symbol

σ

Population Variance symbol

σ²