Find definitions and interpretation guidance for every statistic that is provided with store descriptive statistics.

The mean is the average of the data, which is the sum of all the observations divided by the number of observations.

For example, the wait times (in minutes) of five customers in a bank are: 3, 2, 4, 1, and 2. The mean waiting time is calculated as follows:
On average, a customer waits 2.4 minutes for service at the bank.

Use the mean to describe the sample with a single value that represents the center of the data. Many statistical analyses use the mean as a standard measure of the center of the distribution of the data.

The median and the mean both measure central tendency. But unusual values, called outliers, can affect the median less than they affect the mean. If your data are symmetric, the mean and median are similar.

The standard error of the mean (SE Mean) estimates the variability between sample means that you would obtain if you took repeated samples from the same population. Whereas the standard error of the mean estimates the variability between samples, the standard deviation measures the variability within a single sample.

For example, you have a mean delivery time of 3.80 days, with a standard deviation of 1.43 days, from a random sample of 312 delivery times. These numbers yield a standard error of the mean of 0.08 days (1.43 divided by the square root of 312). If you took multiple random samples of the same size, from the same population, the standard deviation of those different sample means would be around 0.08 days.

Use the standard error of the mean to determine how precisely the sample mean estimates the population mean.

A smaller value of the standard error of the mean indicates a more precise estimate of the population mean. Usually, a larger standard deviation results in a larger standard error of the mean and a less precise estimate of the population mean. A larger sample size results in a smaller standard error of the mean and a more precise estimate of the population mean.

Minitab uses the standard error of the mean to calculate the confidence interval.

The standard deviation is the most common measure of dispersion, or how spread out the data are about the mean. The symbol σ (sigma) is often used to represent the standard deviation of a population, while *s* is used to represent the standard deviation of a sample. Variation that is random or natural to a process is often referred to as noise.

Because the standard deviation is in the same units as the data, it is usually easier to interpret than the variance.

Use the standard deviation to determine how spread out the data are from the mean. A higher standard deviation value indicates greater spread in the data. A good rule of thumb for a normal distribution is that approximately 68% of the values fall within one standard deviation of the mean, 95% of the values fall within two standard deviations, and 99.7% of the values fall within three standard deviations.

The standard deviation can also be used to establish a benchmark for estimating the overall variation of a process.

The variance measures how spread out the data are about their mean. The variance is equal to the standard deviation squared.

The greater the variance, the greater the spread in the data.

Because variance (σ^{2}) is a squared quantity, its units are also squared, which may make the variance difficult to use in practice. The standard deviation is usually easier to interpret because it's in the same units as the data. For example, a sample of waiting times at a bus stop may have a mean of 15 minutes and a variance of 9 minutes^{2}. Because the variance is not in the same units as the data, the variance is often displayed with its square root, the standard deviation. A variance of 9 minutes^{2} is equivalent to a standard deviation of 3 minutes.

The coefficient of variation (denoted as COV) is a measure of spread that describes the variation in the data relative to the mean. The coefficient of variation is adjusted so that the values are on a unitless scale. Because of this adjustment, you can use the coefficient of variation instead of the standard deviation to compare the variation in data that have different units or that have very different means.

The larger the coefficient of variation, the greater the spread in the data.

For example, you are the quality control inspector at a milk bottling plant that bottles small and large containers of milk. You take a sample of each product and observe that the mean volume of the small containers is 1 cup with a standard deviation of 0.08 cup, and the mean volume of the large containers is 1 gallon (16 cups) with a standard deviation of 0.4 cups. Although the standard deviation of the gallon container is five times greater than the standard deviation of the small container, their coefficients of variation support a different conclusion.

The coefficient of variation of the small container is more than three times greater than that of the large container. In other words, although the large container has a greater standard deviation, the small container has much more variability relative to its mean.

Large container | Small container |
---|---|

COV = 100 * 0.4 cups / 16 cups = 2.5 | COV = 100 * 0.08 cups / 1 cup = 8 |

Quartiles are the three values–the first quartile at 25% (Q1), the second quartile at 50% (Q2 or median), and the third quartile at 75% (Q3)–that divide a sample of ordered data into four equal parts.

The first quartile is the 25th percentile and indicates that 25% of the data are less than or equal to this value.

The median is the midpoint of the data set. This midpoint value is the point at which half the observations are above the value and half the observations are below the value. The median is determined by ranking the observations and finding the observation that are at the number [N + 1] / 2 in the ranked order. If the number of observations are even, then the median is the average value of the observations that are ranked at numbers N / 2 and [N / 2] + 1.

The median and the mean both measure central tendency. But unusual values, called outliers, can affect the median less than they affect the mean. If your data are symmetric, the mean and median are similar.

Quartiles are the three values–the first quartile at 25% (Q1), the second quartile at 50% (Q2 or median), and the third quartile at 75% (Q3)–that divide a sample of ordered data into four equal parts.

The third quartile is the 75th percentile and indicates that 75% of the data are less than or equal to this value.

The interquartile range (IQR) is the distance between the first quartile (Q1) and the third quartile (Q3). 50% of the data are within this range.

Use the interquartile range to describe the spread of the data. As the spread of the data increases, the IQR becomes larger.

The mean of the data, without the highest 5% and lowest 5% of the values.

Use the trimmed mean to eliminate the impact of very large or very small values on the mean. When the data contain outliers, the trimmed mean may be a better measure of central tendency than the mean.

The sum is the total of all the data values. The sum is also used in statistical calculations, such as the mean and standard deviation.

The minimum is the smallest data value.

In these data, the minimum is 7.

13 | 17 | 18 | 19 | 12 | 10 | 7 |
9 | 14 |

Use the minimum to identify a possible outlier or a data-entry error. One of the simplest ways to assess the spread of your data is to compare the minimum and maximum. If the minimum value is very low, even when you consider the center, the spread, and the shape of the data, investigate the cause of the extreme value.

The maximum is the largest data value.

In these data, the maximum is 19.

13 | 17 | 18 | 19 |
12 | 10 | 7 | 9 | 14 |

Use the maximum to identify a possible outlier or a data-entry error. One of the simplest ways to assess the spread of your data is to compare the minimum and maximum. If the maximum value is very high, even when you consider the center, the spread, and the shape of the data, investigate the cause of the extreme value.

The range is the difference between the largest and smallest data values in the sample. The range represents the interval that contains all the data values.

Use the range to understand the amount of dispersion in the data. A large range value indicates greater dispersion in the data. A small range value indicates that there is less dispersion in the data. Because the range is calculated using only two data values, it is more useful with small data sets.

The uncorrected sum of squares is the sum of the squares of each value in the column. For example, if the column contains x_{1}, x_{2}, ... , x_{n}, then sum of squares calculates (x_{1}^{2} + x_{2}^{2} + ... + x_{n}^{2}). Unlike the corrected sum of squares, the uncorrected sum of squares includes error. The data values are squared without first subtracting the mean.

Skewness is the extent to which the data are not symmetrical.

Use skewness to help you establish an initial understanding of your data.

Kurtosis indicates how the peak and tails of a distribution differ from the normal distribution.

Use kurtosis to initially understand general characteristics about the distribution of your data.

The MSSD is the mean of the squared successive difference. MSSD is an estimate of variance. One possible use of the MSSD is to test whether a sequence of observations is random. In quality control, a possible use of MSSD is to estimate the variance when the subgroup size = 1.

The number of non-missing values in the sample.

In this example, there are 141 recorded observations.

Total count | N | N* |
---|---|---|

149 | 141 |
8 |

The number of missing values in the sample. The number of missing values refers to cells that contain the missing value symbol *.

In this example, 8 errors occurred during data collection and are recorded as missing values.

Total count | N | NMissing |
---|---|---|

149 | 141 | 8 |

The total number of observations in the column. Use to represent the sum of N missing and N nonmissing.

In this example, there are 141 valid observations and 8 missing values. The count is 149.

Count | N | NMissing |
---|---|---|

149 |
141 | 8 |

Cumulative N is a running total of the number of observations in successive categories. For example, an elementary school records the number of students in grades one through six. The CumN column contains the cumulative count of the student population:

Grade Level | Count | CumN | Calculation |
---|---|---|---|

1 | 49 | 49 | 49 |

2 | 58 | 107 | 49 + 58 |

3 | 52 | 159 | 49 + 58 + 52 |

4 | 60 | 219 | 49 + 58 + 52 + 60 |

5 | 48 | 267 | 49 + 58 + 52 + 60 + 48 |

6 | 55 | 322 | 49 + 58 + 52 + 60 + 48 + 55 |

The percent of observations in each group of the By variable. In the following example, there are four groups: Line 1, Line 2, Line 3, and Line 4.

Group (by variable) | Percent |
---|---|

Line 1 | 16 |

Line 2 | 20 |

Line 3 | 36 |

Line 4 | 28 |

The cumulative percent is the cumulative sum of the percentages for each group of the By variable. In the following example, the By variable has 4 groups: Line 1, Line 2, Line 3, and Line 4.

Group (by variable) | Percent | CumP |
---|---|---|

Line 1 | 16 | 16 |

Line 2 | 20 | 36 |

Line 3 | 36 | 72 |

Line 4 | 28 | 100 |