Chapter 3

Parameter

A characteristic of a population. Any measurable characteristic of a population. Ex: The mean of a population.

Variance

A defect of the range is that it is based on only two values, the maximum and the minimum; it does not take into consideration all of the values. the variance does. -Measures the mean amount by which the values in a population, or sample, vary from their mean. Def: The arithmetic mean of the squared deviations from the mean.

Range formula

Maximum value- minimum value = range

Measures of location

Referred to as average. Purpose is to pinpoint the center of a distribution of data.

Rate of Increase overtime

Second application of the geometric mean is to find an average percentage change over a period of time.

Population Mean

Sum of all the values in the population/ number of values in the population.

The Geometric Mean

Useful in finding the average change of percentages,ratios, indexes, or growth rates over time. -Has a wide application in business and economics because we are often interested in finding the percentage changes in sales, salaries, or economic figures, such as the gross domestic product, which compound or build on each other.

Arithmetic mean of grouped Data

- A mean or a standard deviation from grouped data is an estimate of the corresponding actual values.

The Empirical Rule

For 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; and practically all (99.7%) will lie within plus and minus three standard deviations of the mean.

The Weighted Mean

Is a convenient way to compute the arithmetic mean when there are several observations of the same value.

Sample Standard Deviation

Is used as an estimator of the population standard deviation. As noted previously, the population standard deviation is the square roots of the population variance.

The Relative Positions of the Mean, Median, and Mode

- For any symmetric distribution, the mode, the median, and mean are located at the center and are always equal. - If distribution is nonsymmetrical, or skewed, the relationship among the three measures changes. -A positively skewed distribution, the arithmetic mean is the largest of the three measures. Then the mean, then the mode. -If the distribution is highly skewed, the mean would not be a good measure to use. The median and mode would be more representative. -In a negative skewed, mode, median and then mean.

Chebyshev's Theorem

-Allow us to determine the minimum proportion of the values that lie within a specified number of standard deviations of the mean. -At least three out of every four, or 75%, of the values must lie between the mean plus two standard deviations and the mean minus two standard deviations. This relationship applies regardless of the shape of the distribution. -For any set of observations (sample or population), the proportion of the values that lie within k standard deviations of the mean is at least 1-1/k square. where k is any value greater than 1.

The Median

-The midpoint of values after they have been ordered from the minimum to the maximum values. -For median, the data must be at least an ordinal level of measurement. The major properties of the median are: 1. It is not affected by extremely large or small values.Therefore, the median is a valuable measure of location when such values do occur. 2. It can be computed for ordinal-level data or higher.

Mode

-The value of the observation that appears most frequently. -The mode is especially useful in summarizing nominal-level data. -The mode does have disadvantages, however, that cause it to be used less frequently than the mean or median.

Study of Disperson

A measure of location, such as the mean, median, or mode, only describes the center of the data. It is valuable from that standpoint, but it does not tell us anything about the spread of data.

Sample Mean

A sample from the population to estimate a specific characteristic of the population.

Properties of the Arithmetic Mean

A widely used measure of location. Several important properties: 1.To compute a mean, the data must be measured at the interval or ratio level. 2. All the values are included in computing the mean. 3. The mean is unique. That is, there is only one mean in a set of data. 4. The sum of the deviations of each value from the mean in zero.

Measures of dispersion

Also called variation or the spread. Ex: the spread of a, 40- 100 thousand dollars. Average 80 thousand dollars