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Measures of Central Tendency 

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Measures of central tendency/ measures of central location/ summary statistics seek to describe the presented data by identifying the central position within the given data set. The measures include the mean, median and mode. Variance and standard deviation represent the level of dispersion among the data given. The mean of the sample is given as follows

Where ∑x is the sum of the values given in the data set ,µ is the sample mean and n is the total number of values given
The following is the sum of the data given (∑x)
14.5+ 14.6+ 14.7+14.8+ 14.9+15.3+14.9+15.5+14.8+15.2+ 15+15.1+15+14.4+15.8+14+16+16.1+15.8+14.5+14.6+14.8+14.8+14.6+14.5+14.7+14.9+14+14.2+14.1=4
∑x =446.1
N=30
Mean =446.1/30=14.87
The median is calculated by finding the central value after arranging the values from the smallest to the largest
14, 14, 14.1, 14.2,14.4, 14.5,14.5,14.5, 14.6, 14.6, 14.6 14.7,14.7, 14.8, 14.8,14.8,14.8, 14.9, 14.9,14.9,15,15, 15.1, 15.2, 15.3, 15.5,15.8,15.8, 16, 16.1
Median==14.8
The mode is the value that appears the most number of times in the data set and the value is 14.8. The Value appears four times.



x f (x-mean) f(x-mean)2
14 2 -0.87 1.5138
14.1 1 -0.77 0.5929
14.2 1 -0.67 0.4489
14.4 1 -0.47 0.2209
14.5 3 -0.37 0.1369
14.6 3 -0.27 0.0729
14.7 2 -0.17 0.0578
14.8 4 -0.07 0.0196
14.9 3 0.03 0.027
15 2 0.13 0.0338
15.1 1 0.23 0.0529
15.2 1 0.33 0.1089
15.3 1 0.43 0.1849
15.5 1 0.63 0.3969
15.8 2 0.93 1.7298
16 1 1.13 1.2769
16.1 1 1.23 1.5129

Total= 30 9.0109

The data is from the sample hence the variance is given by the following formula

=9.0109/29=0.3107
Standard deviation is the square root of the variance



The standard deviation of the sample given is 0.5574
The upper and lower limit expresses the highest and lowest value of the mean at a particular confidence interval. It is the range within which the mean can exist at a given level of confidence. The 95 percent confidence interval for the data from the sample is calculated as follows:
Mean=14.87
Se=0.55574
The value of t-critical at 95% significance level = t (29, 0.05) = 0.2052
The Upper limit=Sample Mean plus se. (t-critical)
=14.87+0.5574(0.2052) = 14.9844
The lower limit is sample mean minus the se.(t-critical)
=14.87-0.5574(0.2052) = 14.7556
The 95% confidence interval can be given as:
P [14.76
 

Asked in Standards & Testing - 561 days ago

Tags: Measures of Central Tendency 

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