Exploration: Comparing Two Groups

Measuring Central Tendency

The three most common measures of location are the mean, the median, and the mode. The mode describes a typical measurement in terms of the most common outcome or most frequently occurring score. In Figure 9, the mode of each distribution is the household expenditure value directly under the peak of the curve. A limitation in using the mode is that a distribution can have more than one. This indicates that the mode will not always have a unique value and, thus, cannot be recommended for general use.

In contrast to the mode, the median and mean are always unique values. The median is the middle-most score in a distribution. The median() function is used to find the median of the distribution. The best known and most frequently used measure of central tendency is the mean, or the average. The mean() function is used to find the mean of a distribution.

## Marginal mean and median household per capita expenditure
vlss %>%
  summarize(
    M = mean(expend),
    Med = median(expend)
    )

## # A tibble: 1 x 2
##       M   Med
##   <dbl> <dbl>
## 1  213.  160.

The median household per capita expenditure is $160, and the mean household per capita expenditure is $213. In symmetric distributions, the mean and median can be equal or nearly so. However, in asymmetric distributions, the two can differ, sometimes drastically.

Conditional Means and Medians

The mean and median computed in the previous section summarize the marginal distribution, as area is ignored. Though the marginal estimates are useful, the goal is to compute the conditional estimates of a typical household per capita expenditure for each area. To do this we add a group_by() layer into our piping syntax prior to computing the summary values. Consider the following syntax:

## Condition mean and median household per capita expenditure
vlss %>%
  group_by(area) %>%
  summarize(
    M = mean(expend),
    Med = median(expend)
    )

## # A tibble: 2 x 3
##   area      M   Med
##   <chr> <dbl> <dbl>
## 1 Rural  157.  135.
## 2 Urban  349.  279.

The mean household per capita expenditure for the urban area is more than twice that for the rural area. This is consistent with Figure 9 that shows the urban distribution being right-shifted relative to the rural distribution. This suggests that the average household per capita expenditure differs for urban and rural areas in the sample.

Measuring Variation

When an analysis deals with at least two groups, as in the rural/urban comparisons, it is important to consider group differences in variability and well as location. Variability within the groups influences the evaluation of location differences. High within-group variability can be an overwhelming feature that can render location differences as irrelevant, or at least less relevant. On the other hand, low within-group variability can work to accentuate location differences.

Consider the examples in Figure 10. In both panels the mean difference between the distributions is the same. However, the large within-group variation in the rural distribution in the left-panel makes the interpretation of group differences less clear for these data than for the data shown in the right-panel. In fact, it can be argued that the most important feature is the fact that the urban distribution is almost entirely contained within the rural distributions. This means, for example, that though the rural mean is lower than the urban mean, there are several rural households that are higher than the urban mean, and some that are higher than any urban households.

In contrast, in the right-panel of Figure 10, there is essentially no overlap between the two distributions. This means that the mean difference also characterizes the difference between almost every pair of households from the two distributions. If we were to randomly select one rural and one urban household, the rural household would almost surely have a lower annual income. The same cannot be said of the overlapping distributions in the right-panel of Figure 10.

Figure 10: Simulated density plots for the distribution of household per capita expenditures conditioned on area showing large within-group variation (LEFT PANEL) and small within-group variation (RIGHT PANEL).

Two summary measures of variation—the standard deviation and variance—are based on the deviations of the data from the mean. The sd() and var() functions can be used to compute these quantities, respectively. The syntax below illustrates the use of the functions to find the standard deviation for both the marginal and conditional distributions of household per capita expenditures.

## SD and variance of the marginal distribution of household per capita expenditure
vlss %>%
  summarize(
    SD = sd(expend),
    Variance = var(expend)
    )

## # A tibble: 1 x 2
##      SD Variance
##   <dbl>    <dbl>
## 1  180.   32222.

## SD and variance of the conditional distributions of household per capita expenditure
vlss %>%
  group_by(area) %>%
  summarize(
    SD = sd(expend),
    Variance = var(expend)
  )

## # A tibble: 2 x 3
##   area     SD Variance
##   <chr> <dbl>    <dbl>
## 1 Rural  96.9    9385.
## 2 Urban 250.    62564.

Based on these conditional summaries, the rural households show less variation than the urban households. This is consistent with Figure 9 that shows the urban distribution being wider relative to the rural distribution. The average household per capita expenditure is more homogeneous for rural than for urban households. There are some caveats regarding indexes of variation. Most notably, measures of variation are sensitive to asymmetry, and their values can be inflated by even a single extreme value. For this reason, the skewness of the distributions should be considered when comparing measures of variation computed on such distributions.

Another measure of variation that often gets reported in the educational and behavioral sciences, is the standard error of the mean. The idea underlying the standard error is that different samples drawn from the same population have different values of the sample mean. This is a consequence of random sampling and the fact that sample information is always incomplete relative to the population. The standard error of the mean is the standard deviation of all the possible sample means for a given sample size. As such, this measure offers an indication of the precision of the sample mean, when it is used as an estimate of the population mean. The smaller the standard error the greater the precision. The standard error for the mean is computed as

\[ \mathrm{SE}_{\bar{Y}}=\frac{\mathrm{SD}_Y}{\sqrt{n}}. \]

where \(\mathrm{SD}_Y\) is the standard deviation of the observed measurements on some variable \(Y\). The standard error of the mean is computed for both the urban and rural households in the syntax below. The standard error for the rural group is approximately four times smaller than that of the urban group (\(\frac{6.01}{1.48} \approx 4\)). This suggests that the sample mean for the rural households is a more precise estimate of the rural population mean than the sample urban mean is for the urban population. The use of the sample estimates and standard error for estimating population parameters is discussed further in Chapter 9.

vlss %>%
  group_by(area) %>%
  summarize(
    SD = sd(expend),
    N = n()
    ) %>%
  mutate(
    SE = SD / sqrt(N)
  )

## # A tibble: 2 x 4
##   area     SD     N    SE
##   <chr> <dbl> <int> <dbl>
## 1 Rural  96.9  4269  1.48
## 2 Urban 250.   1730  6.01

Measuring Skewness

Skewness is a numerical measure that helps summarize a distribution’s departure from symmetry about its mean. A completely symmetric distribution has a skewness value of zero.⁶ Positive values suggest a positively skewed (right-tailed) distribution with an asymmetric tail extending toward more positive values, whereas negative values suggest a negatively skewed (left-tailed) distribution with an asymmetric tail extending toward more negative values.

The e1071 package⁷ provides a function called skewness(), which computes the skewness value for a sample distribution based on three common algorithms. This function is supplied with the argument type=2 to compute G1, a slightly modified version of skewness that is a better population estimate (e.g., Joanes & Gill, 1998). The syntax below shows the use of skewness() to find the G1 values for the conditional distributions.

## Skewness (G1) of the conditional distributions of household per capita expenditure
vlss %>%
  group_by(area) %>%
  summarize(
    G1 = skewness(expend, type = 2) 
  )

## # A tibble: 2 x 2
##   area     G1
##   <chr> <dbl>
## 1 Rural  4.28
## 2 Urban  2.73

These values suggest that both the urban and rural distributions are positively skewed, but more so for the rural group. The following guidelines are offered as help in interpreting the skewness statistic. Like all guidelines these should be used with a healthy amount of skepticism. All statistics should be interpreted in terms of the types and purposes of the data analysis, as well as the substantive area of the research.

• If \(G_1=0\), the distribution is symmetric.
• If \(\left|G_1\right| < 1\), the skewness of the distribution is slight.⁸
• If \(1 < \left|G_1\right| < 2\), the the skewness of the distribution is moderate.
• If \(\left|G_1\right| >2\), the distribution is quite skewed.

The above guidelines indicate that both distributions in the example are severely positively skewed. Furthermore, the rural distribution is more asymmetric than the urban distribution. This is again consistent with Figure 9, which shows the rural distribution has a longer tail relative to its mean than the urban distribution. The distribution of rural households shows relatively less density for household per capita expenditures above the mean than below the mean. This asymmetry is even more evident for urban households.

Measuring Kurtosis

Kurtosis is often used as a numerical summarization of the “peakedness” of a distribution, referring to the relative concentration of scores in the center, tail, and shoulders. Normal distributions have a kurtosis value of zero and are called mesokurtic.⁹ Distributions that reflect a more peaked and heavy-tailed distribution than the normal distribution have positive kurtosis values, and are said to be leptokurtic. Distributions which are flatter and lighter-tailed than the normal distribution have negative kurtosis values and are said to be platykurtic. Dyson & Cantab (1943, p. 360) suggest an “amusing mnemonic”—which was attributed to Gossett (Student, 1927)—for the above terms:

Platykurtic curves, like the platypus, are squat with short tails. Leptokurtic curves are high with long tails, like the kangaroo—noted for “lepping”.

The left- and right-hand panels of Figure 11 depict distributions with different kurtosis values. The mesokurtic distribution is shown for a basis of comparison in both figures. The distributions in the left-hand panel show positive kurtosis, whereas the distributions in the right-hand panel show negative kurtosis.

Figure 11: LEFT PANEL: Kernel density estimate for a mesokurtic distribution (dashed, orange line) and a leptokurtic distribution (solid, purple line). The leptokurtic distributions are skinnier and more peaked than the mesokurtic distribution. RIGHT PANEL: Kernel density estimate for a mesokurtic distribution (dashed, orange line) and a platykurtic distribution (solid, blue line). The platykurtic distribution is flatter than the mesokurtic distribution.

The kurtosis() function provided in the e1071 package can be used to compute the sample kurtosis value for a distribution based on three common algorithms. We use this function with the argument type=2 to compute G2, a slightly modified version of the kurtosis statistic that is a better population estimate of kurtosis (e.g., Joanes & Gill, 1998). The syntax below shows the use of kurtosis() to find the G2 values for the conditional distributions.

## Kurtosis (G2) of the conditional distributions of household per capita expenditure
vlss %>%
  group_by(area) %>%
  summarize(
    G2 = kurtosis(expend, type = 2) 
  )

## # A tibble: 2 x 2
##   area     G2
##   <chr> <dbl>
## 1 Rural  42.7
## 2 Urban  14.0

The kurtosis statistics for the conditional distributions suggest that both distributions are severely leptokurtic indicating that these distributions are more peaked than a normal distribution. They also have more density in the tails of the distribution than we would expect to see in a normal distribution. One can see in Figure 9 that the rural distribution is even more peaked than the urban distribution.

While the kurtosis statistic is often examined and reported by educational and behavioral scientists who want to numerically describe their samples, it should be noted that “there seems to be no universal agreement about the meaning and interpretation of kurtosis” (Moors, 1986, p. 283). Most textbooks in the social sciences describe kurtosis in terms of peakedness and tail weight. Balanda & MacGillivray (1988, p. 116) define kurtosis as “the location- and scale free movement of probability mass from the shoulders of a distribution into its center and tails … peakedness and tail weight are best viewed as components of kurtosis. Other statisticians have suggested that it is a measure of the bimodality present in a distribution (e.g., Darlington, 1970; Finucan, 1964). Perhaps it is best defined by Mosteller & Tukey (1977), who suggest that like location, variation, and skewness, kurtosis should be viewed as a”vague concept" that can be formalized in a variety of ways.