5.1.4. Sample Statistics
The Variable Selection Dialogue for this procedure offers three types of data to analyse (see 5.0.2. OneSample Data Types). A text box is also provided on this dialogue to enter the size of the total population from which the sample is drawn. The default value of 0 means that the total population is not known and the program assumes an infinite population. A nonzero population value affects only the standard error of mean in output.
The Output variables in rows check box allows you to transpose the output matrix. This will be useful when you wish to use the output from this procedure (such as means and standard errors) for further analysis in other procedures.
Output for different data options differ slightly. For instance, the grouped data output includes the Sheppard’s correction for the second and fourth moments but it does not include minimum, maximum and range.
For ungrouped data, the method used in computing the median, lower and upper quartiles is indicated in the output. This can be one of the six methods described in the previous section 5.1.3.1. Quantile Methods.
The following statistics can be calculated for ungrouped (option 1) and frequency and grouped data (options 2 and 3). Let n be the number of valid observations (i.e. excluding missing values) and f_{i} the frequency of data point X_{i} given in column 2. Note that for ungrouped data f_{i} = 1, i = 1, …, n.
Size: Number of cases (rows) in the sample, including missing values.
Missing: Number of missing cases in the sample. In frequency and grouped data a case is considered missing when either or both of value and frequency are missing.
Total Frequency:
N = n for ungrouped data.
Mean: The weighted arithmetic mean is:
Geometric Mean: The weighted geometric mean is:
Harmonic Mean: The weighted harmonic mean is:
The following relationship should hold if X_{i} ≥ 0, i = 1, …, n:
Median: For ungrouped data, this is computed using the quantile method selected in step two of the Quantiles (Percentiles) procedure, as described in section 5.1.3.1. Quantile Methods.
For frequency and grouped data, both value and frequency columns are sorted in ascending order according to values. For frequency data, half of total frequency is found and the median is calculated as above. For grouped data, median is calculated by interpolation as:
where:
· L is the lower class boundary of the class containing the median,
· the summation term is the sum of frequencies of all classes lower than the median class,
· C is the size of median class interval and
· N is the total frequency as defined above.
Lower Quartile: Calculations are similar to that of median, except for 25% quantile instead of 50%.
Upper Quartile: Calculations are similar to that of median, except for 75% quantile instead of 50%.
Interquartile Range: Difference between upper and lower quartiles.
Minimum: Smallest observed value in data (not available for grouped data).
Maximum: Greatest observed value in data (not available for grouped data).
Range: Difference between maximum and minimum values (not available for grouped data).
Sum: The weighted sum is:
Sum of Squares: The weighted sum of squares is:
Root Mean Square (Quadratic mean):
Unbiased Variance:
Unbiased Standard Deviation:
Standard Error of Mean:
Standard Error with Finite Population Correction: Available only when total population is known and it is greater than the total frequency.
Coefficient of Variation:
Variance:
Standard Deviation:
Sheppard’s Correction for 2^{nd} Moment (Variance): Available for only grouped data:
where C is the size of uniform class interval.
Mean Deviation:
3rd Moment About the Mean:
4^{th} Moment About the Mean:
Unbiased 3^{rd} Moment:
Sheppard’s Correction for the 4^{th} Moment: Available for only grouped data:
where C is the size of uniform class interval.
Moment Coefficient of Skewness:
An alternative definition of skewness is given in section 5.1.1. Summary Statistics.
Moment Coefficient of Kurtosis:
An alternative definition of kurtosis is given in section 5.1.1. Summary Statistics.
Pearson’s Second Coefficient of Skewness:
Example 1: Ungrouped data
Open PARTEST and select Statistics 1 → Descriptive Statistics → Sample Statistics. Select Haemoglobin, Platelets, log Leucocytes, and Systolic BP (C10 to C13) as [Variable]s, uncheck the Output variables in rows box and click [Finish].
Sample Statistics
Quantile Method: Simple Average

Haemoglobin 
Platelets 
log Leucocytes 
Systolic BP 
Size 
10.0000 
10.0000 
10.0000 
10.0000 
Missing 
0.0000 
0.0000 
0.0000 
0.0000 
Mean 
0.5300 
0.0300 
0.5900 
3.1000 
Geometric Mean 
* 
* 
* 
* 
Harmonic Mean 
* 
* 
* 
* 
Median 
0.6000 
0.1000 
0.6500 
2.0000 
Lower Quartile 
1.5000 
1.0000 
1.6000 
2.0000 
Upper Quartile 
0.0000 
0.6000 
0.9000 
8.0000 
Interquartile Range 
1.5000 
1.6000 
2.5000 
10.0000 
Minimum 
2.4000 
2.2000 
3.2000 
6.0000 
Maximum 
2.3000 
1.9000 
1.7000 
14.0000 
Range 
4.7000 
4.1000 
4.9000 
20.0000 
Sum 
5.3000 
0.3000 
5.9000 
31.0000 
Sum of Squares 
22.0700 
13.3900 
25.1700 
437.0000 
Root Mean Square 
1.4856 
1.1572 
1.5865 
6.6106 
Unbiased Variance 
2.1401 
1.4868 
2.4099 
37.8778 
Unbiased Standard Deviation 
1.4629 
1.2193 
1.5524 
6.1545 
Standard Error of Mean 
0.4626 
0.3856 
0.4909 
1.9462 
Coefficient of Variation 
2.6186 
38.5588 
2.4961 
1.8834 
Variance 
1.9261 
1.3381 
2.1689 
34.0900 
Standard Deviation 
1.3878 
1.1568 
1.4727 
5.8387 
Mean Deviation 
1.1500 
0.9020 
1.2500 
4.9200 
3rd Moment About Mean 
1.3179 
0.4318 
0.1544 
69.6720 
4th Moment About Mean 
9.2971 
4.2938 
9.5652 
2527.7857 
Unbiased 3rd Moment 
1.8304 
0.5998 
0.2144 
96.7667 
Moment Coefficient of Skewness 
0.4930 
0.2790 
0.0483 
0.3500 
Moment Coefficient of Kurtosis 
2.5060 
2.3981 
2.0334 
2.1751 
Pearson’s Skewness Coefficient 
0.1513 
0.3371 
0.1222 
0.5652 
Example 2: Variables in rows
Continuing from the last example, go back to Variable Selection Dialogue, check the Output variables in rows box and click [Next]. From the Output Options Dialogue select only the last three options and click [Finish].
Sample Statistics

Moment Coefficient of Skewness 
Moment Coefficient of Kurtosis 
Pearson’s Skewness Coefficient 
Haemoglobin 
0.4930 
2.5060 
0.1513 
Platelets 
0.2790 
2.3981 
0.3371 
log Leucocytes 
0.0483 
2.0334 
0.1222 
Systolic BP 
0.3500 
2.1751 
0.5652 
Example 3: Frequency data
Open TIMESER, select Statistics 1 → Descriptive Statistics → Sample Statistics and select the second data option Column 1 contains Data and Column 2 contains Frequencies. Select Surface Area (C13) as [Column 1] and Blemishes (C14) as [Column 2] and enter 150 in the Total Population box. The following results are obtained:
Sample Statistics
Surface Area: contains data, Blemishes contains frequencies

Surface Area 
Size 
20.0000 
Missing 
0.0000 
Total Frequency 
94.0000 
Total Population 
150.0000 
Mean 
0.8462 
Geometric Mean 
0.8265 
Harmonic Mean 
0.8070 
… 
… 
Root Mean Square 
0.8653 
Unbiased Variance 
0.0330 
Unbiased Standard Deviation 
0.1817 
Standard Error of Mean 
0.0187 
Standard Error with Finite Population 
0.0115 
Coefficient of Variation 
0.2136 
Variance 
0.0327 
Standard Deviation 
0.1807 
Mean Deviation 
0.1443 
3rd Moment About Mean 
0.0004 
4th Moment About Mean 
0.0017 
Unbiased 3rd Moment 
0.0004 
Moment Coefficient of Skewness 
0.0635 
Moment Coefficient of Kurtosis 
1.6210 
Pearson’s Skewness Coefficient 
0.1024 