Chapter 3

Rounding Rule for the Mean

The mean should be rounded to one more decimal place than occurs in the raw data. For example, if the raw data are given in whole numbers, the mean should be rounded to the nearest tenth. If the data are given in tenths, the mean should be rounded to the nearest hundredth, and so on.

Population standard deviation

The square root of the variance. The symbol for the population standard deviation is σ. The corresponding formula for the population standard deviation is:

Coefficient of Variation

The standard deviation divided by the mean with the result expressed as a percentage. For Samples: CVar = s/x * 100 For Populations: CVar = o(greek o)/mue * 100

Distribution Shapes

The three most important shapes are positively skewed, symmetric, and negatively skewed

A parameter

a characteristic or measure obtained by using all the data values from a specific population.

A Statistic:

a characteristic or measure obtained by using the data values from a sample. *General Rounding Rule*the basic rounding rule is that when computations are done in the calculation, rounding should not be done until the final answer is calculated.

symmetric

a distribution in which the data values are uniformly distributed about the mean

positively skewed

a distribution in which the majority of the data values fall to the left of the mean

negatively skewed

a distribution in which the majority of the data values fall to the right of the mean

The sample mean

denoted by (pronounced "X bar"), is calculated by using sample data. The sample mean is a statistic. Xbar= X1+X2+X3+.........=E(Greek Letter)/n: where n represents the total number of values in the sample.

The population mean

denoted by μ (pronounced "mew"), is calculated by using all the values in the population. The population mean is a parameter. μ=X1+X2+X3+............../N=E(Greek Letter)X/N: where N represents the total number of values in the population. **In statistics, Greek letters are used to denote parameters, and Roman letters are used to denote statistics. Assume that the data are obtained from samples unless otherwise specified.**

Percentiles

divide the data set into 100 equal groups.

Range:

is the highest value minus the lowest value. The symbol R is used for the range. Make sure the range is given as a single number R = Highest Value - Lowest Value

The Mode:

is the value that occurs most often in the data set. It is sometimes said to be the most typical case.

Population variance

the average of the squares of the distance each value is from the mean. The symbol for the population variance is σ2 (σ is the Greek lowercase letter sigma). In formula: X = individual value, mue = population mean, N = population size

The median

the midpoint of the data array. The symbol for the median is MD.

Properties and Uses of Central Tendency: Mean

1. The mean is found by using all the values of the data. 2. The mean varies less than the median or mode when samples are taken from the same population and all three measures are computed for these samples. 3. The mean is used in computing other statistics, such as the variance. 4. The mean for the data set is unique and not necessarily one of the data values. 5. The mean cannot be computed for the data in a frequency distribution that has an open-ended class. 6. The mean is affected by extremely high or low values, called outliers, and may not be the appropriate average to use in these situations.

Properties and Uses of Central Tendency: Median

1. The median is used to find the center or middle value of a data set. 2. The median is used when it is necessary to find out whether the data values fall into the upper half or lower half of the distribution. 3. The median is used for an open-ended distribution. 4. The median is affected less than the mean by extremely high or extremely low values.

Properties and Uses of Central Tendency: Midrange

1. The midrange is easy to compute. 2. The midrange gives the midpoint. 3. The midrange is affected by extremely high or low values in a data set.

z score or standard score

A value is obtained by subtracting the mean from the value and dividing the result by the standard deviation. The symbol for a standard score is z. The formula is:

The Mean

Also known as the arithmetic average, is found by adding the values of the data and dividing by the total number of values. For example, the mean of 3, 2, 6, 5, and 4 is found by adding 3 + 2 + 6 + 5 + 4 = 20 and dividing by 5; hence, the mean of the data is 20 ÷ 5 = 4. The values of the data are represented by X's. In this data set, X1 = 3, X2 = 2, X3 = 6, X4 = 5, and X5 = 4. To show a sum of the total X values, the symbol ∑ (the capital Greek letter sigma) is used, and ∑X means to find the sum of the X values in the data set. **Do not round up during this step!**

Finding the Median

Step 1 Arrange the data values in ascending order. Step 2 Determine the number of values in the data set. Step 3 a) If n is odd, select the middle data value as the median. b) If n is even, find the mean of the two middle values. That is, add them and divide the sum by 2.

Modal Class:

Is the class with the largest frequency.

Chebyshev's Theorem

It tells the proportion of values from a dataset that will fall within "k" standard deviations of the mean

The Midrange

The midrange is a rough estimate of the middle. It is found by adding the lowest and highest values in the data set and dividing by 2. It is a very rough estimate of the average and can be affected by one extremely high or low value. **Do Not Round Up**