Descriptive Statistics

Descriptive Statistics. These statistics are used to describe a population or a sample.
  1. Population Mean: () and Sample Average (x, or [x bar])


  2. Standard Deviation: measurement of the spread in individual data points to reflect the uncertainty of a single measurement.

    1. Population standard deviation (σ). For large sample sets (usually more than 20 measurements) or when the population mean () is known.

    2. Sample standard deviation (s). For small sample sets (usually less than 20 measurements) when the sample average ([x bar]) is used.

    3. Pooled standard deviation (spooled ). When several small sets have the same sources of indeterminate error (ie: the same type of measurement but different samples) the standard deviations of the individual data sets may be pooled to more accurately determine the standard deviation of the analysis method.


  3. Standard Error of the Meanm). The standard error of the mean is the uncertainty in the average. This is different from the standard deviation (σ), which is the variation for each individual measurement. Notice that when N is 1 (a single measurement) σm =σ .

    1. If [σ] is known, the uncertainty in the mean is:

    2. If [σ] is unknown, use the t-score to compensate for the uncertainty in s. The value for t is obtained from a table for appropriate % confidence level and for N-1 degrees of freedom. (N-1 because one degree of freedom is used to calculate the mean.) Since the uncertainty is a range that could be greater or less than the mean, a two-sided value should be used for t.


  4. z-Score. Normalizes data points so that the average is 0 and the standard deviation is 1. The cumulative normal distribution (z-score) shows what percentage of a normal distribution is bounded by a given value for z. One sided distributions, the distribution is the area from -&inf; and z. Two sided values give the area between z. To illustrate this:

    1. For a 1 sided distribution 97.72% of all data points will be less than 2 standard deviations above the average.

    2. For a 2 sided distribution 68.28% of all data points will be between 1 standard deviation from the average.

    Cumulative Normal Distribution.
    The area under a gaussian distribution where z is the population standard deviation (σ)
    z 0 1 2 3
    p1sided 0.500 0.8414 0.9772 0.9986
    p2sided 0.00 0.6828 0.9544 0.9876


  5. Confidence Interval. The confidence interval is the preferred method for describing the range of uncertainty in a value. The confidence interval is expressed as a range of uncertainties at a stated percent confidence. This percent confidence reflects the percent certainty that the value is within the stated range.

    1. If the population standard deviation (σ) is known. The standard error of the mean (σm) combined with the z-score (from a table for the desired Confidence Level) is used to express the uncertainty in the mean as a range. This is the confidence interval at the stated certainty. The percentage used should always be stated. This method is widely used to report results with a percent certainty and is expressed as follows:

    2. If the population standard deviation (σ) is unknown. The sample standard deviation (s) may be used to estimate the confidence interval. This is the preferred method for reporting the uncertainty in experimental results. It takes into account the number of measurements made, the variance in the measurements, and expresses the range at the stated percent confidence level.

    3. Based upon the confidence interval calculated above, an experimental result should be expressed as:

      5.3 1.2 at the 95% confidence level


Student's t

P1 sided t.60 t.70 t.80 t.90 t.95 t.975 t.99 t.995
P2 sided t.20 t.40 t.60 t.80 t.90 t.95 t.98 t.99
df
1 0.325 0.727 1.376 3.078 6.314 12.71 31.82 63.66
2 0.289 0.617 1.061 1.886 2.920 4.303 6.965 9.925
3 0.277 0.584 0.978 1.638 2.353 3.182 4.541 5.841
4 0.271 0.569 0.941 1.533 2.132 2.776 3.747 4.607
5 0.267 0.559 0.920 1.476 2.015 2.571 3.365 4.032
6 0.265 0.553 0.920 1.440 1.943 2.447 3.143 3.707
7 0.263 0.549 0.896 1.415 1.895 2.365 2.998 3.499
8 0.262 0.546 0.889 1.397 1.860 2.306 2.896 3.355
9 0.261 0.543 0.883 1.383 1.833 2.262 2.821 3.250
10 0.260 0.542 0.879 1.372 1.812 2.228 2.764 3.169
-
20 0.257 0.533 0.860 1.325 1.725 2.086 2.528 2.845
-
inf 0.283 0.524 0.842 1.282 1.645 1.960 2.326 2.576