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Understanding Population & Sample Statistics: Mean, Median, Dispersion & Confidence Interv, Exercises of Statistics

University of North Florida (UNF)Statistics

An in-depth exploration of statistics, focusing on population and sample mean, median, measures of location and dispersion, and confidence intervals. It covers the philosophical and practical meanings of a population, statistical samples, strategies for defining survey objectives, and various measures of location and dispersion such as mean, median, population variance, sample variance, and standard deviation. Additionally, it discusses point and interval estimates, sampling distributions, and the Central Limit Theorem.

What you will learn

  • How is the standard error of the mean calculated?
  • What is the difference between population variance and sample variance?
  • How is the median calculated?
  • What is the difference between population and sample mean?
  • What is the Central Limit Theorem and how does it apply to statistics?

Typology: Exercises

2021/2022

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Basic Statistical Concepts
Statistical Population
• The entire underlying set of observations
from which samples are drawn.
– Philosophical meaning: all observations that could
ever be taken for range of inference
• e.g. all barnacle populations that have ever existed, that
exist or that will exist
– Practical meaning: all observations within a
reasonable range of inference
• e.g. barnacle populations on that stretch of coast
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Download Understanding Population & Sample Statistics: Mean, Median, Dispersion & Confidence Interv and more Exercises Statistics in PDF only on Docsity!

Basic Statistical Concepts

Statistical Population

  • The entire underlying set of observations from which samples are drawn. - Philosophical meaning: all observations that could ever be taken for range of inference - e.g. all barnacle populations that have ever existed, that exist or that will exist - Practical meaning: all observations within a reasonable range of inference - e.g. barnacle populations on that stretch of coast

Statistical Sample

  • A representative subset of a population.
    • What counts as being representative
      • Unbiased and hopefully precise

Strategies

  • Define survey objectives: what is the goal of survey or experiment? What are you hypotheses?
  • Define population parameters to estimate (e.g. number of individuals, growth, color etc).
  • Implement sampling strategy
    • measure every individual (think of implications in terms of cost, time, practicality especially if destructive)
    • measure a representative portion of the population (a sample)
  • Population mean ( ) - the average value
  • Sample mean = estimates 
  • Population median - the middle value
  • Sample median estimates population median
  • In a normal distribution the mean=median (also the mode), this is not ensured in other distributions

y

Y Y Mean & median Median Mean

Measures of location

Measures of dispersion

  • Population variance ( ^2 ) - average sum of squared deviations from mean
  • Measured sample variance ( s^2 ) estimates population variance
  • Standard deviation ( s )
    • square root of variance
    • same units as original variable

( xi - x ) 2 n - 1

( xi - ) 2 n

( xi - x ) 2 n - 1

( xi - x ) 2 n - 1

Measures (statistics) of Dispersion

Population variance ^2 =

Sample variance s^2 =

Sample standard deviation s = (^) 

  • Note, units are squared
  • Denominator is (n)
  • Note, units are squared
  • Denominator is (n-1)
  • Note, units are not squared

Population Sum of Squares ( xi - ) 2

Sample Sum of Squares SS = ( xi - x ) 2

s^2 n

s x

( xi - x ) ( yi - y ) n - 1

More Statistics of Dispersion

Standard error of the mean s (^) x =

Coefficient of variation CV =

Covariance s (^) xy =

  • This is also the Standard Deviation^  of the sample means
  • Measurement of variation independent of units
  • Expressed as a percentage of mean
  • Measure of how two variables covary
  • Range is between - and +
  • Value depends in part on range in data
  • bigger numbers yield bigger values of covariance

=  n

s  n

s

y

P(y)

y

y

P(y)

Sampling distribution of sample means

Multiple samples

  • multiple sample means

True Mean = 25

22 27

12 33 25

41

31

23

(^3619)

Mean = 21.

23 24

36

28 28 25 21 17 40 16 Mean = 25.

Means 21.

25.

Estimate of Mean

Number of cases

10 20 30 40

Sampling distribution of mean

  • The sampling distribution of the sample mean approaches a normal distribution as n gets larger - Central Limit Theorem.
  • The mean of this sampling distribution is , the mean of original population.

Estimate of Mean ( x)

Probability

Estimate of Mean

(^015 20 25 30 )

4

8

12

16

of cases

0.3 (^) Proportion per Bar

Large number of Samples

~2 SEM^ ~2 SEM

Probability2.5% 2.5%

Estimate of Mean ( x)

Standard deviation can be calculated for any distribution The standard deviation of the distribution of sample means can be calculated the same as for a given sample



( x (^) i - x ) 2 However: N - 1 To do so would require an immense sampling effort, hence an approximation is used:

Where: s = sample standard deviation and n = number of replicates in the sample

 n

s  n

s s x ~^ SEM =

s (^) x =

x

Standard error of mean

  • population SD estimated by sample SE:

s / n

  • measures precision of sample mean
  • how close sample mean is likely to be to

true population mean

Standard error of mean

  • If SE is low:
    • repeated samples would produce similar sample means
    • therefore, any single sample mean likely to be close to population mean
  • If SE is high:
    • repeated samples would produce very different sample means
    • therefore, any single sample mean may not be close to population mean

0.00 (^0 10) Estimate of Mean 20 30 40

Probability

0.00 (^0 10) Estimate of Mean 20 30 40

Probability

1 SEM=2 1 SEM=

Effect of Standard error on estimate of (assume df= large)

 

~2 SEM ~2 SEM

~2 SEM ~2 SEM 2.5%^ 2.5%

Distribution of sample means

Calculate the proportion of sample means within a range of values.

Transform distribution of means to a distribution with mean = 0 and standard deviation = 1

95%

99%

P( (^) y ) (^) y

t statistic

s / n

y ^ 

0.0 -5 -4 -3 -2 -1 0 1 2 3 4 5

Probability

s / n

y ^ 

t =

Null distribution

t statistic – interpretation and

units

  • The deviation between the sample and population mean is expressed in terms of Standard error (i.e. Standard deviations of the sampling distribution)
  • Hence the value of t’s are in standard errors
  • For example t=2 indicates that the deviation ( y-) is equal to 2 x the standard error

s / n

y ^ 

Degrees of Freedom .01 .02 .05 .10. 1 63.66 31.82 12.71 6.314 3. 2 9.925 6.965 4.303 2.920 1. 3 5.841 4.541 3.182 2.353 1. 4 4.604 3.747 2.776 2.132 1. 5 4.032 3.365 2.571 2.015 1. 10 3.169 2.764 2.228 1.812 1. 15 2.947 2.602 2.132 1.753 1. 20 2.845 2.528 2.086 1.725 1. 25 2.787 2.485 2.060 1.708 1. z 2.575 2.326 1.960 1.645 1.

Two tailed t-values

Probability

Probabilities of occurring outside the range

  • t (^) df to + t (^) df

-5 -4 -3 -2 -1-5 -4 -3 -2 -1 00 11 22 33 44 55

-2.78 95%+2.

4 df

s / n t =y ^ 

s / n

t =y ^ 

Degrees of Freedom .005/.01 .01/.02 .025/.05 .05/.10 .10/. 1 63.66 31.82 12.71 6.314 3. 2 9.925 6.965 4.303 2.920 1. 3 5.841 4.541 3.182 2.353 1. 4 4.604 3.747 2.776 2.132 1. 5 4.032 3.365 2.571 2.015 1. 10 3.169 2.764 2.228 1.812 1. 15 2.947 2.602 2.132 1.753 1. 20 2.845 2.528 2.086 1.725 1. 25 2.787 2.485 2.060 1.708 1. z 2.575 2.326 1.960 1.645 1.

-5 -4 -3 -2 -1-5 -4 -3 -2 -1 00 11 22 33 44 55

-2.78 95%+2. -5 -4 -3 -2 -1-5 -4 -3 -2 -1 00 11 22 33 44 55

95%+2. -5 -4 -3 -2 -1-5 -4 -3 -2 -1 00 11 22 33 44 55

-2.132 95%

One and two tailed t-values (df 4)

2 tailed 1 tailed 1 tailed

s / n

t =y ^ 

The t statistic

  • This t statistic follows a t -distribution, which has a mathematical formula.
  • Same as normal distribution for n >30 otherwise flatter, more spread than normal distribution.
  • Different t distributions for different sample sizes < 30 (actually df which is n -1).
  • The proportions of t values between particular t values, yield a confidence estimate (the likelihood that the true mean is in the range)

For n = 5 (df = 4), 95% of all t values occur between t = -2.78 and t = +2.

95%

Pr( t )

-2.78 0 +2. t

  • Probability is 95% that t is between -2.78 and +2.
  • Probability is 95% that is between -2.78 and +2.
  • Rearrange equation to solve for

s n

y  

-5 -4 -3 -2 -1-5 -4 -3 -2 -1 00 11 22 33 44 55

-2.78 95%+2.

Worked example (Lovett et al. 2000) Sample mean 61. Sample SD 5. SE 0.

  • The t value (95%, 38df) = 2.02 (from a t -table)
  • 2.5% of t values are greater than 2.
  • 2.5% of t values are less than -2.
  • 95% of t values are between -2.02 and +2.

P {61.92 - 2.02 (5.24 / 39) <  < 61.92 + 2.02 (5.24 / 39)} = 0.

P {60.22 <  < 63.62} = 0.

Degrees of Freedom .01 .02 .05 .10. 1 63.66 31.82 12.71 6.314 3. 2 9.925 6.965 4.303 2.920 1. 3 5.841 4.541 3.182 2.353 1. 4 4.604 3.747 2.776 2.132 1. 5 4.032 3.365 2.571 2.015 1. 10 3.169 2.764 2.228 1.812 1. 15 2.947 2.602 2.132 1.753 1. 20 2.845 2.528 2.086 1.725 1. 25 2.787 2.485 2.060 1.708 1. 38 2.705 2.426 2.020 1.685 1.

Confidence Interval (2 tailed) assume 95% CI is desired

Probability

95%

Lovett et al 38 df. (2000)

yt ( s / n ) yt ( s / n ) 61.92 – 2.02(0.84) 60.

61.92 + 2.02(0.84) <  < 63.

Pr[yt ( s / n )  y^  t ( s / n ) ]

Sample mean 61. SEM 0. DF 38

  • The interval 60.22 – 63.62 will contain  95% of the time.
  • We are 95% confident that the interval 60.
    • 63.62 contains .

Effect on Confidence Interval

Case Mean Samplesize (SS) Standarddeviation (SD)

StandardError Probability(%) LowerConfidence Limit

UpperConfidence Limit Reference 61.92 39 5.24 0.834 95% 60.22 63. Double SD

61.92 39 10.48 1.68 95% 58.53 65. Reduce SS

61.92 20 5.24 1.17 95% 59.47 64. Increase %

61.92 39 5.24 0.834 99% 59.65 64.