Estimate Probability When You Only Know Population Mean and Standard Deviation

ESTIMATING THE POPULATION Mean

C. Patrick Doncaster

Whenever y'all collect a sample of measurements, you will want to summarise its defining characteristics. If the data are approximately normally distributed around some central tendency, and many types of biological data are, then three parametric statistics tin can provide much of the essential data. The sample hateful, , tells you what is the boilerplate measurement from your sample; the standard difference (SD) tells you how much variation there is in the in the data around the sample mean; the standard error (SE) indicates the uncertainty associated with viewing the sample mean equally an guess of the hateful of the whole population, .

Parameter		Description	Instance
1.	Variable	A belongings that varies in a measurable style between subjects in a sample.	Weight of seeds of the Princess Edible bean Phaseolus vulgaris (in: Samuels, M.Fifty. 1991. Statistics for the Life Sciences. Macmillan).
2.	Sample	A collection of individual observations selected by a specified process. In most cases the sample size is given by the number of subjects (i.e. each is measured in one case but).	A sample of 25 Princess Bean seeds, selected at random from the full production of an arable field.	WEIGHT (mg) 343,755,431,480,516,469,69 4,659,441,562,597,502,612, 549,348,469,545,728,416,53 6,581,433,583,570,334
3.	Sample mean	The sum of all observations in the sample, divided past the size of the sample, N. The sample mean is an estimate of the population mean, ("mu") which is one of 2 parameters defining the normal distribution (the other is , see below).	The sample mean This comes from a population, the total production of the field, which follows a normal distribution and has a hateful = 500 mg.
iv.	Sum of squares, SS	The squared distance between each information betoken (Y_i ) and the sample mean, summed for all Northward information points.	The sample sums of squares
5.	Variance, v,	The variance in a commonly distributed population is described by the average of North squared deviations from the mean. Variance usually refers to a sample, withal, in which example information technology is calculated every bit the sum of squares divided by Northward-1 rather than N.	The sample variance 5 = SS / (N - one) = 12,928
6.	Sample standard deviation, SD, south	Describes the dispersion of information about the mean. It is equal to the square root of the variance. For a large sample size, = , and the standard difference of the sample approaches the population standard deviation, ("sigma"). It is and so a property of the normal distribution that 95% of observations will lie inside ane.960 standard deviations of the mean, and 99% inside .	The sample standard divergence south = = 113.7 mg The standard difference of the population from which the sample was fatigued is = 120 mg.
7.	Normal distribution	A bell-shaped frequency distribution of a continuous variable. The formula for the normal distribution contains two parameters: the mean, giving its location, and the standard deviation, giving the shape of the symmetrical 'bell'. This distribution arises commonly in nature when myriad independent forces, themselves subject to variation, combine additively to produce a central tendency. Many parametric statistics are based on the normal distribution because of this, and also its holding of describing both the location (mean) and dispersion (standard deviation) of the data. Since dispersion is measured in squared deviations from the hateful, it can exist partitioned between sources, permitting the testing of statistical models.	The weights of Princess Bean seeds in the population follows a normal distribution (shown in the graph, with frequency on the horizontal centrality). Some 95% of the seeds are within 1.96 standard deviations of the mean, which is = 500 235 mg.
8.	Standard error of the hateful, SE	Describes the dubiousness, due to sampling error, in the mean of the information. Information technology is calculated by dividing the standard departure by the square root of the sample size ( ), and then information technology gets smaller as the sample size gets bigger. In other words, with a very big N, the sample mean approaches the population mean. If random samples of Due north measurements were taken from whatsoever population (not necessarily normal) with hateful and standard deviation , the hateful of the sampling distribution of would equal the population mean . Moreover, the standard deviation of sample means around the population hateful would be given by .	The standard error of the mean
9.	Confidence interval for	Regardless of the underlying distribution of data, the sample means from repeated random samples of size due north would have a distribution that approached normal for large n, with 95% of sample ways at . With only one sample mean and standard mistake SE, these can withal be taken as best estimates of the parametric hateful g and standard deviation of sample means. It is then possible to compute 95% confidence limits for at (for large sample sizes). For small sample sizes, The 95% conviction limits for are computed at .	The 95% confidence intervals for k from the sample of 25 Princess Bean seeds are at: = 526.1 2.069 22.74 = 526.1 47.05. The sample is thus representative of the population hateful, which we happen to know is 500 mg. If nosotros didn't know this, the sample would nonetheless lead us to accept a null hypothesis that the population mean lies anywhere between 479.05 and 573.15 mg.