Question

Calculate the value of commonly used statistics to find measure of spread for the runs scored by the Indian Cricket Team based on their scores in last 15 One Day Internationals while batting first. The data are shown in Table \(3.14 .\) $$ \begin{array}{l} \text { Table 3.14 Data for Problem } 3.5\\\ \begin{array}{l|l|l|l|l|l|l|l|l|l|l|l|l|l|l} \hline 281 & 307 & 251 & 429 & 241 & 189 & 256 & 194 & 267 & 385 & 228 & 299 & 247 & 331 & 389 \\ \hline \end{array} \end{array} $$

Short answer

The Range, Mean, Median, Mode, Variance, and Standard Deviation calculated provide the statistical measures of spread for the runs scored.

Step-by-step solution

Arrange the Data

First of all, the data needs to be organized in increasing order for easier calculations.

Calculate the Range

The range is the difference between the largest and smallest data points.

Calculate the Mean

The mean is calculated by summing all data points and dividing the total by the number of data points.

Find the Median

The median is the data point at the exact middle of a data set. If the dataset has an even number of observations, the median will be the average of the two middle numbers.

Find the Mode

The mode is the most frequently occurring data point in a data set. If no data point is repeated, then there is no mode.

Calculate Variance

Variance reflects the diversity of the data points. It's calculated by the average of the squared differences from the Mean.

Calculate Standard Deviation

The standard deviation is calculated by taking the square root of the variance, providing a measure of the amount of variation or dispersion in the data set.

Key concepts

Range in Statistics

The range is one of the simplest measures of spread in statistics, assessing the extent of variation in a data set. To calculate the range, you subtract the smallest value from the largest value in the data set.

For example, if an Indian cricket team's scores are {189, 241, 251, ..., 429}, the range is calculated as: Range = Maximum score - Minimum score = 429 - 189 = 240. This result indicates that the spread of scores, from the lowest to the highest, is 240 runs.

Mean Calculation

The mean, or average, is a fundamental statistical concept used to find the central value of a data set. To compute the mean, add up all the numerical values and divide by the count of the numbers.

For the given cricket scores, the mean is found by summing all the scores and dividing by 15 (the number of matches). If the total sum of the scores is 4673, the calculation would be: Mean = Sum of scores / Number of matches = 4673 / 15 = 311.53. The mean score for the team while batting first, therefore, is approximately 311 runs.

Median in Statistics

The median is the middle value of a data set when the values are arranged in ascending or descending order. If there is an even number of observations, the median is the average of the two middle numbers.

In our cricket team's data, once arranged, if the 8th and 9th values are 256 and 267, the median would be: Median = (256 + 267) / 2 = 261.5. This means that half of the team's scores are below 261.5 and half are above.

Mode in Data Set

The mode is the value that appears most frequently in a data set. A set may have one mode, more than one mode, or no mode at all.

In case of cricket scores like {189, 229, ..., 307, 307, ...}, the score 307 appears most frequently. Therefore, 307 is the mode of this data set. If no score is repeated, the data set is considered to have no mode. Identifying the mode helps understand the consistency of performance in a team's scores.

Variance Calculation

Variance measures how much the data points in a set diverge from the mean. It's calculated by averaging the squared differences of each data point from the mean.

For precise computation, you would follow these steps:

1. Compute the mean of the data.
2. Subtract the mean from each data point and square the result.
3. Average these squared differences.

If the mean of our cricket scores is around 311, the variance calculation involves squaring the differences between each score and 311, then averaging those squared numbers. A high variance indicates a wide spread of scores, suggesting inconsistency in performance.

Standard Deviation

The standard deviation is a measure of the dispersion of a set of data from its mean. It's the square root of the variance. A lower standard deviation indicates that the data points tend to be close to the mean, while a higher standard deviation indicates that the data are spread out over a wider range.

Following the variance calculation, if our variance is 1445.7, the standard deviation will be: Standard Deviation = Square root of Variance = √1445.7 ≈ 38.02. This can be interpreted as the 'average' distance from the mean and speaks to how variable the cricket team's performance is in terms of the runs they score.