Читать книгу Introduction to Statistical Process Control - Muhammad Amir Aslam - Страница 20

The weights of 10 items selected from the assembly line are given as

24.3, 24.9, 23.6, 25.1, 26.7, 22.0, 21.8, 23.4, 24.1, and 20.8. Calculate median of the weights.

Arranging the values in ascending order

20.8, 21.8, 22.0, 23.4, 23.6, 24.1, 24.3, 24.9, 25.1, and 26.7.

The median for grouped may be defined as

where l is the lower class boundary of the median class and c is the cumulative frequency.

The family of median consists of partitioning of observations into different parts. As median is defined as the value that divides the data into 2 equal parts, the observations may be divided into 4, 10, or 100 equal parts. These measures are known as the quantiles. When data are divided into four parts, then it is called quartiles. Upper and lower quartiles or Q₁ and Q₃ may be defined as

and

The decile is defined as the value that divides the data into 10 equal parts.

The other values of deciles can be calculated accordingly as the subscript is multiplied with the numerator. For example, the sixth decile may be calculated as

The percentiles may be defined as the value that divides the data into 100 equal parts:

The other values of percentiles can be calculated accordingly as the subscript is multiplied with the numerator. For example, the 70th percentile may be calculated as

The formula for the grouped data can be constructed accordingly.

 iii) The Mode

The mode is defined as the most repeated value in any data set. It is denoted by . If there is no repetition in the data, then there will be no value of this measure. This is the single measurement whose results may be no value of mode, one value of mode, two values of the mode, or more values of mode in any data set.

The mode for the grouped data may be calculated as

 iv) The Range

The range is defined as the difference between the maximum value and the minimum value of the data set:

or

The range for the grouped data may be calculated as

 v) The Quartile Deviation

The quartile deviation is defined as the half of the difference between the third and the first quartiles of the data set. The formula for the quartile deviation is defined as

It is to be noted that all the measures of dispersion are positive.

The coefficient of QD can be calculated as

 vi) The Mean Deviation

The mean deviation is defined as the average of the deviations from the mean or median; the deviations are taken without algebraic sign. So the average deviation calculated from mean is known as the mean deviation from mean and is defined as

And the mean deviation from median and is defined as

And for the grouped data these measures can be defined as

And the mean deviation from median and is defined as

The coefficient of MD is defined as

It is to be noted that coefficient of median deviation is smaller than the coefficient of mean deviation.

 vii) The Variance

The variance is defined as the average of the squared deviation from mean. It is denoted by S². The formula for the variance may be defined as

The variance for the grouped data may be defined as

 viii) The Standard Deviation

The standard deviation may be defined as the positive square root of the variance. It is denoted by S. The formula for the standard deviation may be defined as

The standard deviation for the grouped data may be defined as

The coefficient of variation may be defined as