The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations

The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations.

3-1© 2010 Pearson Prentice Hall. All rights reserved

The population arithmetic mean is computed using all the individuals in a population.

The population mean is a parameter.

The population arithmetic mean is denoted by .


The sample arithmetic mean is computed using sample data.

The sample mean is a statistic.

The sample arithmetic mean is denoted by .

x


If x1, x2, …, xN are the N observations of a variable from a population, then the population mean, µ, is

1 2 Nx x x

N


If x1, x2, …, xn are the n observations of a variable from a sample, then the sample mean, , is

1 2 nx x xx

n

x


The median of a variable is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median.



A numerical summary of data is said to be resistant if extreme values (very large or small) relative to the data do not affect its value substantially.



The mode of a variable is the most frequent observation of the variable that occurs in the data set.

If there is no observation that occurs with the most frequency, we say the data has no mode.


Tally data to determine most frequent observation


The range, R, of a variable is the difference between the largest data value and the smallest data values. That is

Range = R = Largest Data Value – Smallest Data Value


The population variance of a variable is the sum of squared deviations about the population mean divided by the number of observations in the population, N.

That is it is the mean of the sum of the squared deviations about the population mean.


The population variance is symbolically represented by σ2 (lower case Greek sigma squared).

Note: When using the above formula, do not round until the last computation. Use as many decimals as allowed by your calculator in order to avoid round off errors.


The Computational Formula


The sample variance is computed by determining the sum of squared deviations about the sample mean and then dividing this result by n – 1.


Note: Whenever a statistic consistently overestimates or underestimates a parameter, it is called biased. To obtain an unbiased estimate of the population variance, we divide the sum of the squared deviations about the mean by n - 1.


The population standard deviation is denoted by

It is obtained by taking the square root of the population variance, so that

The sample standard deviation is denoted by

s

It is obtained by taking the square root of the sample variance, so that

2s s3-18© 2010 Pearson Prentice Hall. All rights reserved



(a) Compute the population mean and standard deviation.

(b) Draw a histogram to verify the data is bell-shaped.

(c) Determine the percentage of patients that have serum HDL within 3 standard deviations of the mean according to the Empirical Rule.

(d) Determine the percentage of patients that have serum HDL between 34 and 69.1 according to the Empirical Rule. (e) Determine the actual percentage of patients that have serum HDL between 34 and 69.1.


(a) Using a TI-83 plus graphing calculator, we find

(b)

7.11 and 4.57


22.3 34.0 45.7 57.4 69.1 80.8 92.5

(e) 45 out of the 54 or 83.3% of the patients have a serum HDL between 34.0 and 69.1.

(c) According to the Empirical Rule, 99.7% of the patients that have serum HDL within 3 standard deviations of the mean.

(d) 13.5% + 34% + 34% = 81.5% of patients will have a serum HDL between 34.0 and 69.1 according to the Empirical Rule.







The kth percentile, denoted, Pk, of a set of data is a value such that k percent of the observations are less than or equal to the value.


Quartiles divide data sets into fourths, or four equal parts.

• The 1st quartile, denoted Q1, divides the bottom 25% the data from the top 75%. Therefore, the 1st quartile is equivalent to the 25th percentile.

• The 2nd quartile divides the bottom 50% of the data from the top 50% of the data, so that the 2nd quartile is equivalent to the 50th percentile, which is equivalent to the median.

• The 3rd quartile divides the bottom 75% of the data from the top 25% of the data, so that the 3rd quartile is equivalent to the 75th percentile.







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The arithmetic mean of a variable is computed by determining the sum of all the values of the variable in the data set divided by the number of observations