BU306-Chapter 3

Measures of location

The purpose of the measures of location are to pinpoint the center of a distribution of data. The arithmetic mean The median The mode

Mode

The value of the observation that appears most frequently 65 66 71* 71* 71* 76 76 80

Finding the mean of grouped data

This happens if we only have access to data that is in a frequency distribution. You have the classes, the frequency's, now we need to find the midpoint of the classes. When we find the midpoint, we will multiply by the frequencies and add and divide by the total number of frequency's.

Formula for finding the mean of grouped data

x̄ = ∑fM / n x̄ is the mean of the sample M is the midpoint of each class f is the frequency in each class n is the total number of frequencies

Sample mean formula

x̄ = ∑x / n x̄ represents the sample mean. "x bar" n is the number of values in the sample x represents any particular value ∑ is "sigma" for adding or sum ∑x is the sum of the x values in the population

Arithmetic mean

The most widely used measure of location. It requires interval scale level of data.

Properties of the arithmetic mean

- Every set of interval and ratio-level data has a mean. - All the values in the data set are included in computing the mean. - The mean is unique. - The sum of the deviations of each value from the mean is zero. This important property is explained on the next slide.

Properties of the median

- There is a unique median for each data set. - It is not affected by extremely large or small values and is therefore a valuable measure of central tendency when such values occur. - It only requires ordinal-level data - It can be computed for an open-ended frequency distribution as long as the median does not lie in an open-ended class.

Computing the variance

1. Take each value and subtract the mean 2. Square each of those deviations 3. Sum up all the squared deviations 4. Divide the sum by N (the number of observations in the population)

Parameter

A measurable characteristic of a population

Dispersion

A measure of location, such as the mean or the median, only describes the center of the data but it does not tell us anything about the spread of the data. How spread out the values of the data set are from the mean.

Even amount of numbers when trying to figure the median

Add the 2 middle numbers and divide by 2.

Standard deviation of grouped data

After we go through the process to find the mean of the grouped data, we can find the standard deviation by squaring each deviation then multiplying by the frequency

Sample variance

Almost the same as population variance except we divide by n-1 instead of N

Empirical rule

For ONLY a symmetrical, bell-shaped frequency distribution, approximately 68% of the observations will lie within plus and minus one standard deviation of the mean; about 95% of the observations will lie within plus and minus two standard deviations of the mean ex: We have a mean of 100 and a standard deviation of 10. 100-10=90 and 100+10=110. About 68% of my data will lie between 90 and 110.

Sample mean

For ungrouped data, it is the sum of all the sample values divided by the number of sample values

Population mean

For ungrouped data, this is the sum of all the population values divided by the total number of population values. µ = ∑x / N µ represents the population mean N is the number of values in the population x represents any particular value ∑ is "sigma" for adding or sum ∑x is the sum of the x values in the population

Standard deviation formula

Just the square root of the variance √o~^2 o~^2 is the population variance

Variance

The arithmetic mean of the squared deviations from the mean.

Negatively skewed data

Skewed to the left; the mean, is less than the median, which is less than the mode.

Positively skewed data

Skewed to the right; There are a few larger numbers that are pulling the mean to the right. In positively skewed data the mean, is greater than the median, which is greater than the mode.

Range

The highest value minus the lowest value. Give an idea of the span of the data

Median

The midpoint of the values after they have been ordered from the minimum to the maximum values. Literally the number in the middle of the data, once it has been ordered from smallest to largest.

Standard deviation

The most common measure of dispersion. The square root of the variance (population and sample variance).

Sample variance formula

s^2 = ∑(x- x̄)^2 / n-1 s^2 is the sample variance x is the value of each observation in the sample x̄ is the mean of the sample n is the number of observations in the sample

Population variance formula

o~ ^2 = ∑ (x - µ)2 / N o~ ^2 is the population variance x is the value of a particular observation in the population µ is the arithmetic mean of the population N is the number of observation in the population

Measures of dispersion

range, variance, standard deviation

Formula for finding the standard deviation of grouped data

s=√∑f(M- x̄)^2 / n-1